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NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry - In the Class 12 Maths Chapter 11 NCERT solutions you will learn how to use vector algebra to study three dimensional geometry. Understanding vector algebra is a prerequisite for 3d geometry class 12. The practice of three dimensional geometry class 12 solutions will be helpful in solving three dimensional problems given in NCERT Books for Class 12.
NCERT Class 12 Maths solutions chapter 11 makes the study simple and effective. It is also very helpful to solve problems asked in the CBSE Board as well as entrance exam related to chapter three dimensional geometry class 12. NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry will build your base for many other higher-level concepts like tensors and manifolds due to that this ch 11maths class 12 becomes very important. If you want to know more about 3d geometry class 12 then you can also check NCERT solutions for other classes.
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>> Distance Formula:
The distance between two points A(x1, y1, z1) and B(x2, y2, z2) is given by:
AB = √[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
The distance between a point A(x, y, z) and the origin O(0, 0, 0) is given by:
OA = √(x² + y² + z²)
>> Section Formula: The coordinates of the point R, which divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) internally or externally in the ratio m:n, are given by:
Internal Division: (mx2 + nx1) / (m + n), (my2 + ny1) / (m + n), (mz2 + nz1) / (m + n)
External Division: (mx2 - nx1) / (m - n), (my2 - ny1) / (m - n), (mz2 - nz1) / (m - n)
>> Midpoint Formula: The coordinates of the mid-point of the line segment joining (x1, y1) and (x2, y2) are:
[(x1 + x2) / 2, (y1 + y2) / 2]
>> Coordinates of Centroid of a Triangle: Given vertices (x1, y1), (x2, y2), and (x3, y3) of a triangle, the coordinates of the centroid are:
[(x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3]
>> Incentre of a Triangle: Given vertices (x1, y1), (x2, y2), and (x3, y3) of a triangle, the coordinates of the incenter are:
[(ax1 + bx2 + cx3) / (a + b + c), (ay1 + by2 + cy3) / (a + b + c)]
>> Centroid of a Tetrahedron: Given vertices (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4) of a tetrahedron, the coordinates of the centroid are:
[(x1 + x2 + x3 + x4) / 4, (y1 + y2 + y3 + y4) / 4, (z1 + z2 + z3 + z4) / 4]
>> Direction Cosines of a Line: If a directed line OP makes angles α, β, and γ with the positive X-axis, Y-axis, and Z-axis, respectively, then the direction cosines l, m, and n are:
l = cos α, m = cos β, n = cos γ Also, the sum of squares of direction cosines is always 1:
l² + m² + n² = 1
>> Direction Ratios of a Line: Direction ratios of a line are denoted as a, b, and c. They are proportional to the direction cosines:
l/a = m/b = n/c
>> Angle between Two Line Segments: If a1, b1, c1, and a2, b2, c2 are the direction ratios of two lines and θ is the acute angle between them, then:
cos θ = |(a1a2 + b1b2 + c1c2) / (√(a1² + b1² + c1²) √(a2² + b2² + c2²))|
>> Perpendicular and Parallel Lines: Two lines are perpendicular if: a1a2 + b1b2 + c1c2 = 0
Two lines are parallel if: a1/a2 = b1/b2 = c1/c2
>> Projection of a Line Segment on a Line: Given points P(x1, y1, z1) and Q(x2, y2, z2) and a line with direction cosines l, m, n, the projection of PQ on the line is:
|l(x2 - x1) + m(y2 - y1) + n(z2 - z1)|
>> Equation of a Plane: A plane in 3-D space can be represented in various forms:
General form: ax + by + cz + d = 0 (where a, b, c are not all zero)
Normal form: lx + my + nz = p
Plane through a point (x1, y1, z1): a(x - x1) + b(y - y1) + c(z - z1) = 0
Intercept form: (x/a) + (y/b) + (z/c) = 1
Vector form: (r- a).n = 0 or r.n = a.n
>> Planes Parallel to Axes: Planes parallel to the X-axis, Y-axis, and Z-axis are represented as:
Plane Parallel to X-axis: by + cz + d = 0
Plane Parallel to Y-axis: ax + cz + d = 0
Plane Parallel to Z-axis: ax + by + d = 0
Free download NCERT Class 12 Maths Chapter 11 Question Answer for CBSE Exam.
NCERT Class 12 Maths Chapter 11 Question Answer - Exercise: 11.1
Question:1 If a line makes angles with the x, y and z-axes respectively, find its direction cosines.
Answer:
Let the direction cosines of the line be l,m, and n.
So, we have
Therefore the direction cosines of the lines are .
Question:2 Find the direction cosines of a line which makes equal angles with the coordinate axes.
Answer:
If the line is making equal angle with the coordinate axes. Then,
Let the common angle made is with each coordinate axes.
Therefore, we can write;
And as we know the relation;
or
Thus the direction cosines of the line are
Question:3 If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?
Answer:
GIven a line has direction ratios of -18, 12, – 4 then its direction cosines are;
Line having direction ratio -18 has direction cosine:
Line having direction ratio 12 has direction cosine:
Line having direction ratio -4 has direction cosine:
Thus, the direction cosines are .
Question:4 Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.
Answer:
We have the points, A (2, 3, 4),B (– 1, – 2, 1),C (5, 8, 7);
And as we can find the direction ratios of the line joining the points is given by
The direction ratios of AB are i.e.,
The direction ratios of BC are i.e., .
We can see that the direction ratios of AB and BC are proportional to each other and is -2 times.
AB is parallel to BC. and as point B is common to both AB and BC,
Hence the points A, B and C are collinear.
Answer:
Given vertices of the triangle (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).
Finding each side direction ratios;
Direction ratios of side AB are i.e.,
Therefore its direction cosines values are;
SImilarly for side BC;
Direction ratios of side BC are i.e.,
Therefore its direction cosines values are;
Direction ratios of side CA are i.e.,
Therefore its direction cosines values are;
NCERT Class 12 Maths Chapter 11 Question Answer - Exercise: 11.2
Question:1 Show that the three lines with direction cosines
Answer:
GIven direction cosines of the three lines;
And we know that two lines with direction cosines and are perpendicular to each other, if
Hence we will check each pair of lines:
Lines ;
the lines are perpendicular.
Lines ;
the lines are perpendicular.
Lines ;
the lines are perpendicular.
Thus, we have all lines are mutually perpendicular to each other.
Answer:
We have given points where the line is passing through it;
Consider the line joining the points (1, – 1, 2) and (3, 4, – 2) is AB and line joining the points (0, 3, 2) and (3, 5, 6).is CD.
So, we will find the direction ratios of the lines AB and CD;
Direction ratios of AB are
or
Direction ratios of CD are
or .
Now, lines AB and CD will be perpendicular to each other if
Therefore, AB and CD are perpendicular to each other.
Answer:
We have given points where the line is passing through it;
Consider the line joining the points (4, 7, 8) and (2, 3, 4) is AB and line joining the points (– 1, – 2, 1) and (1, 2, 5)..is CD.
So, we will find the direction ratios of the lines AB and CD;
Direction ratios of AB are
or
Direction ratios of CD are
or .
Now, lines AB and CD will be parallel to each other if
Therefore we have now;
Hence we can say that AB is parallel to CD.
Answer:
It is given that the line is passing through A (1, 2, 3) and is parallel to the vector
We can easily find the equation of the line which passes through the point A and is parallel to the vector by the known relation;
, where is a constant.
So, we have now,
Thus the required equation of the line.
Answer:
Given that the line is passing through the point with position vector and is in the direction of the line .
And we know the equation of the line which passes through the point with the position vector and parallel to the vector is given by the equation,
So, this is the required equation of the line in the vector form.
Eliminating , from the above equation we obtain the equation in the Cartesian form :
Hence this is the required equation of the line in Cartesian form.
Answer:
Given a line which passes through the point (– 2, 4, – 5) and is parallel to the line given by the ;
The direction ratios of the line, are 3,5 and 6 .
So, the required line is parallel to the above line.
Therefore we can take direction ratios of the required line as 3k , 5k , and 6k , where k is a non-zero constant.
And we know that the equation of line passing through the point and with direction ratios a, b, c is written by: .
Therefore we have the equation of the required line:
or
The required line equation.
Question:7 The cartesian equation of a line is . Write its vector form .
Answer:
Given the Cartesian equation of the line;
Here the given line is passing through the point .
So, we can write the position vector of this point as;
And the direction ratios of the line are 3 , 7 , and 2.
This implies that the given line is in the direction of the vector, .
Now, we can easily find the required equation of line:
As we know that the line passing through the position vector and in the direction of the vector is given by the relation,
So, we get the equation.
This is the required equation of the line in the vector form.
Answer:
GIven that the line is passing through the and
Thus the required line passes through the origin.
its position vector is given by,
So, the direction ratios of the line through and are,
The line is parallel to the vector given by the equation,
Therefore the equation of the line passing through the point with position vector and parallel to is given by;
Now, the equation of the line through the point and the direction ratios a, b, c is given by;
Therefore the equation of the required line in the Cartesian form will be;
OR
Answer:
Let the line passing through the points and is AB;
Then as AB passes through through A so, we can write its position vector as;
Then direction ratios of PQ are given by,
Therefore the equation of the vector in the direction of AB is given by,
We have then the equation of line AB in vector form is given by,
So, the equation of AB in Cartesian form is;
or
Question:10 Find the angle between the following pairs of lines:
Answer:
To find the angle A between the pair of lines we have the formula;
We have two lines :
and
The given lines are parallel to the vectors ;
where and respectively,
Then we have
and
Therefore we have;
or
Question:10 Find the angle between the following pairs of lines:
Answer:
To find the angle A between the pair of lines we have the formula;
We have two lines :
and
The given lines are parallel to the vectors ;
where and respectively,
Then we have
and
Therefore we have;
or
Question:11 Find the angle between the following pair of lines:
Answer:
Given lines are;
and
So, we two vectors which are parallel to the pair of above lines respectively.
and
To find the angle A between the pair of lines we have the formula;
Then we have
and
Therefore we have;
or
Question:11 Find the angle between the following pair of lines:
Answer:
Given lines are;
and
So, we two vectors which are parallel to the pair of above lines respectively.
and
To find the angle A between the pair of lines we have the formula;
Then we have
and
Therefore we have;
or
Question:12 Find the values of p so that the lines and are at right angles.
Answer:
First we have to write the given equation of lines in the standard form;
and
Then we have the direction ratios of the above lines as;
and respectively..
Two lines with direction ratios and are perpendicular to each other if,
Thus, the value of p is .
Question:13 Show that the lines and are perpendicular to each other.
Answer:
First, we have to write the given equation of lines in the standard form;
and
Then we have the direction ratios of the above lines as;
and respectively..
Two lines with direction ratios and are perpendicular to each other if,
Therefore the two lines are perpendicular to each other.
Question:14 Find the shortest distance between the lines
Answer:
So given equation of lines;
and in the vector form.
Now, we can find the shortest distance between the lines and , is given by the formula,
Now comparing the values from the equation, we obtain
Then calculating
So, substituting the values now in the formula above we get;
Therefore, the shortest distance between the two lines is units.
Question:15 Find the shortest distance between the lines
Answer:
We have given two lines:
and
Calculating the shortest distance between the two lines,
and
by the formula
Now, comparing the given equations, we obtain
Then calculating determinant
Now calculating the denominator,
So, we will substitute all the values in the formula above to obtain,
Since distance is always non-negative, the distance between the given lines is
units.
Question:16 Find the shortest distance between the lines whose vector equations are and
Answer:
Given two equations of line
in the vector form.
So, we will apply the distance formula for knowing the distance between two lines and
After comparing the given equations, we obtain
Then calculating the determinant value numerator.
That implies,
Now, after substituting the value in the above formula we get,
Therefore, is the shortest distance between the two given lines.
Question:17 Find the shortest distance between the lines whose vector equations are
Answer:
Given two equations of the line
in the vector form.
So, we will apply the distance formula for knowing the distance between two lines and
After comparing the given equations, we obtain
Then calculating the determinant value numerator.
That implies,
Now, after substituting the value in the above formula we get,
Therefore, units are the shortest distance between the two given lines.
NCERT class 12 three dimensional geometry ncert solutions - Exercise: 11.3
Question:1(a) In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
Answer:
Equation of plane Z=2, i.e.
The direction ratio of normal is 0,0,1
Divide equation by 1 from both side
We get,
Hence, direction cosins are 0,0,1.
The distance of the plane from the origin is 2.
Question:1(b) In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
Answer:
Given the equation of the plane is or we can write
So, the direction ratios of normal from the above equation are, .
Therefore
Then dividing both sides of the plane equation by , we get
So, this is the form of the plane, where are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.
The direction cosines of the given line are and the distance of the plane from the origin is units.
Question:1(c) In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
Answer:
Given the equation of plane is
So, the direction ratios of normal from the above equation are, .
Therefore
Then dividing both sides of the plane equation by , we get
So, this is the form of the plane, where are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.
The direction cosines of the given line are and the distance of the plane from the origin is units.
Question:1(d) In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
Answer:
Given the equation of plane is or we can write
So, the direction ratios of normal from the above equation are, .
Therefore
Then dividing both sides of the plane equation by , we get
So, this is the form of the plane, where are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.
The direction cosines of the given line are and the distance of the plane from the origin is units.
Answer:
We have given the distance between the plane and origin equal to 7 units and normal to the vector .
So, it is known that the equation of the plane with position vector is given by, the relation,
, where d is the distance of the plane from the origin.
Calculating ;
is the vector equation of the required plane.
Question:3(a) Find the Cartesian equation of the following planes:
Answer:
Given the equation of the plane
So we have to find the Cartesian equation,
Any point on this plane will satisfy the equation and its position vector given by,
Hence we have,
Or,
Therefore this is the required Cartesian equation of the plane.
Question:3(b) Find the Cartesian equation of the following planes:
Answer:
Given the equation of plane
So we have to find the Cartesian equation,
Any point on this plane will satisfy the equation and its position vector given by,
Hence we have,
Or,
Therefore this is the required Cartesian equation of the plane.
Question:3(c) Find the Cartesian equation of the following planes:
Answer:
Given the equation of plane
So we have to find the Cartesian equation,
Any point on this plane will satisfy the equation and its position vector given by,
Hence we have,
Or,
Therefore this is the required Cartesian equation of the plane.
Question:4(a) In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
Answer:
Let the coordinates of the foot of perpendicular P from the origin to the plane be
Given a plane equation ,
Or,
The direction ratios of the normal of the plane are 2, 3 and 4 .
Therefore
So, now dividing both sides of the equation by we will obtain,
This equation is similar to where, are the directions cosines of normal to the plane and d is the distance of normal from the origin.
Then finding the coordinates of the foot of the perpendicular are given by .
The coordinates of the foot of the perpendicular are;
or
Question:4(b) In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
Answer:
Let the coordinates of the foot of perpendicular P from the origin to the plane be
Given a plane equation ,
Or,
The direction ratios of the normal of the plane are 0,3 and 4 .
Therefore
So, now dividing both sides of the equation by we will obtain,
This equation is similar to where, are the directions cosines of normal to the plane and d is the distance of normal from the origin.
Then finding the coordinates of the foot of the perpendicular are given by .
The coordinates of the foot of the perpendicular are;
or
Question:4(c) In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
Answer:
Let the coordinates of the foot of perpendicular P from the origin to the plane be
Given plane equation .
The direction ratios of the normal of the plane are 1,1 and 1 .
Therefore
So, now dividing both sides of the equation by we will obtain,
This equation is similar to where, are the directions cosines of normal to the plane and d is the distance of normal from the origin.
Then finding the coordinates of the foot of the perpendicular are given by .
The coordinates of the foot of the perpendicular are;
or ..
Question: 4(d) In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
Answer:
Let the coordinates of the foot of perpendicular P from the origin to the plane be
Given plane equation .
or written as
The direction ratios of the normal of the plane are 0, -5 and 0 .
Therefore
So, now dividing both sides of the equation by we will obtain,
This equation is similar to where, are the directions cosines of normal to the plane and d is the distance of normal from the origin.
Then finding the coordinates of the foot of the perpendicular are given by .
The coordinates of the foot of the perpendicular are;
or .
Answer:
Given the point and the normal vector which is perpendicular to the plane is
The position vector of point A is
So, the vector equation of the plane would be given by,
Or
where is the position vector of any arbitrary point in the plane.
Therefore, the equation we get,
or
So, this is the required Cartesian equation of the plane.
Question:5(b) Find the vector and cartesian equations of the planes
that passes through the point (1,4, 6) and the normal vector to the plane is .
Answer:
Given the point and the normal vector which is perpendicular to the plane is
The position vector of point A is
So, the vector equation of the plane would be given by,
Or
where is the position vector of any arbitrary point in the plane.
Therefore, the equation we get,
So, this is the required Cartesian equation of the plane.
Question:6(a) Find the equations of the planes that passes through three points.
(1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3)
Answer:
The equation of the plane which passes through the three points is given by;
Determinant method,
Or,
Here, these three points A, B, C are collinear points.
Hence there will be an infinite number of planes possible which passing through the given points.
Question:6(b) Find the equations of the planes that passes through three points.
(1, 1, 0), (1, 2, 1), (– 2, 2, – 1)
Answer:
The equation of the plane which passes through the three points is given by;
Determinant method,
As determinant value is not equal to zero hence there must be a plane that passes through the points A, B, and C.
Finding the equation of the plane through the points,
After substituting the values in the determinant we get,
So, this is the required Cartesian equation of the plane.
Question:7 Find the intercepts cut off by the plane 2x + y – z = 5.
Answer:
Given plane
We have to find the intercepts that this plane would make so,
Making it look like intercept form first:
By dividing both sides of the equation by 5 (as we have to make the R.H.S =1) , we get then,
So, as we know that from the equation of a plane in intercept form, where a,b,c are the intercepts cut off by the plane at x,y, and z-axes respectively.
Therefore after comparison, we get the values of a,b, and c.
.
Hence the intercepts are .
Question:8 Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.
Answer:
Given that the plane is parallel to the ZOX plane.
So, we have the equation of plane ZOX as .
And an intercept of 3 on the y-axis
Intercept form of a plane given by;
So, here the plane would be parallel to the x and z-axes both.
we have any plane parallel to it is of the form, .
Equation of the plane required is .
Answer:
The equation of any plane through the intersection of the planes,
Can be written in the form of; , where
So, the plane passes through the point , will satisfy the above equation.
That implies
Now, substituting the value of in the equation above we get the final equation of the plane;
is the required equation of the plane.
Answer:
Here and
and and
Hence, using the relation , we get
or ..............(1)
where, is some real number.
Taking , we get
or
or .............(2)
Given that the plane passes through the point , it must satisfy (2), i.e.,
or
Putting the values of in (1), we get
or
or
which is the required vector equation of the plane.
Answer:
The equation of the plane through the intersection of the given two planes, and is given in Cartesian form as;
or ..................(1)
So, the direction ratios of (1) plane are which are .
Then, the plane in equation (1) is perpendicular to whose direction ratios are .
As planes are perpendicular then,
we get,
or
or
Then we will substitute the values of in the equation (1), we get
or
This is the required equation of the plane.
Question:12 Find the angle between the planes whose vector equations are and .
Answer:
Given two vector equations of plane
and .
Here, and
The formula for finding the angle between two planes,
.............................(1)
and
Now, we can substitute the values in the angle formula (1) to get,
or
or
7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
Answer:
Two planes
whose direction ratios are and whose direction ratios are ,
are said to Parallel:
If,
and Perpendicular:
If,
And the angle between is given by the relation,
So, given two planes
Here,
and
So, applying each condition to check:
Parallel check:
Clearly, the given planes are NOT parallel.
Perpendicular check:
.
Clearly, the given planes are NOT perpendicular.
Then find the angle between them,
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
Answer:
Two planes
whose direction ratios are and whose direction ratios are ,
are said to Parallel:
If,
and Perpendicular:
If,
And the angle between is given by the relation,
So, given two planes
Here,
and
So, applying each condition to check:
Perpendicular check:
.
Thus, the given planes are perpendicular to each other.
2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
Answer:
Two planes
whose direction ratios are and whose direction ratios are ,
are said to Parallel:
If,
and Perpendicular:
If,
And the angle between is given by the relation,
So, given two planes
Here,
and
So, applying each condition to check:
Parallel check:
Thus, the given planes are parallel as
2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0
Answer:
Two planes
whose direction ratios are and whose direction ratios are ,
are said to Parallel:
If,
and Perpendicular:
If,
And the angle between is given by the relation,
So, given two planes
Here,
and
So, applying each condition to check:
Parallel check:
Therefore
Thus, the given planes are parallel to each other.
4x + 8y + z – 8 = 0 and y + z – 4 = 0
Answer:
Two planes
whose direction ratios are and whose direction ratios are ,
are said to Parallel:
If,
and Perpendicular:
If,
And the angle between is given by the relation,
So, given two planes
Here,
and
So, applying each condition to check:
Parallel check:
Clearly, the given planes are NOT parallel as .
Perpendicular check:
.
Clearly, the given planes are NOT perpendicular.
Then finding the angle between them,
Question:14 In the following cases, find the distance of each of the given points from the corresponding given plane
Answer:
We know that the distance between a point and a plane is given by,
.......................(1)
So, calculating for each case;
(a) Point and Plane
Therefore,
(b) Point and Plane
Therefore,
(c) Point and Plane
Therefore,
(d) Point and Plane
Therefore,
NCERT class 12 three dimensional geometry ncert solutions - Miscellaneous Exercise
Answer:
We can assume the line joining the origin, be OA where and the point and PQ be the line joining the points and .
Then the direction ratios of the line OA will be and that of line PQ will be
So to check whether line OA is perpendicular to line PQ then,
Applying the relation we know,
Therefore OA is perpendicular to line PQ.
Answer:
Given that are the direction cosines of two mutually perpendicular lines.
Therefore, we have the relation:
.........................(1)
.............(2)
Now, let us assume be the new direction cosines of the lines which are perpendicular to the line with direction cosines.
Therefore we have,
Or,
......(3)
So, l,m,n are the direction cosines of the line.
where, ........................(4)
Then we know that,
So, from the equation (1) and (2) we have,
Therefore, ..(5)
Now, we will substitute the values from the equation (4) and (5) in equation (3), to get
Therefore we have the direction cosines of the required line as;
Question:3 Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.
Answer:
Given direction ratios and .
Thus the angle between the lines A is given by;
a
Thus, the angle between the lines is
Question:4 Find the equation of a line parallel to x-axis and passing through the origin.
Answer:
Equation of a line parallel to the x-axis and passing through the origin is itself x-axis .
So, let A be a point on the x-axis.
Therefore, the coordinates of A are given by , where .
Now, the direction ratios of OA are
So, the equation of OA is given by,
or
Thus, the equation of the line parallel to the x-axis and passing through origin is
Answer:
Direction ratios of AB are
and Direction ratios of CD are
So, it can be noticed that,
Therefore, AB is parallel to CD.
Thus, we can easily say the angle between AB and CD which is either .
Question:6 If the lines and are perpendicular, find the value of k.
Answer:
Given both lines are perpendicular so we have the relation;
For the two lines whose direction ratios are known,
We have the direction ratios of the lines, and are and respectively.
Therefore applying the formula,
or
For, the lines are perpendicular.
Question:7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane
Answer:
Given that the plane is passing through the point so, the position vector of the point A is and perpendicular to the plane whose direction ratios are and the normal vector is
So, the equation of a line passing through a point and perpendicular to the given plane is given by,
, where
Question:8 Find the equation of the plane passing through (a, b, c) and parallel to the plane .
Answer:
Given that the plane is passing through and is parallel to the plane
So, we have
The position vector of the point is,
and any plane which is parallel to the plane, is of the form,
. .......................(1)
Therefore the equation we get,
Or,
So, now substituting the value of in equation (1), we get
.................(2)
So, this is the required equation of the plane .
Now, substituting in equation (2), we get
Or,
Question:9 Find the shortest distance between lines and .
Answer:
Given lines are;
and
So, we can find the shortest distance between two lines and by the formula,
...........................(1)
Now, we have from the comparisons of the given equations of lines.
So,
and
Now, substituting all values in equation (3) we get,
Hence the shortest distance between the two given lines is 9 units.
Question:10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane.
Answer:
We know that the equation of the line that passes through the points and is given by the relation;
and the line passing through the points,
And any point on the line is of the form .
So, the equation of the YZ plane is
Since the line passes through YZ- plane,
we have then,
or and
So, therefore the required point is
Question : 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.
Answer:
We know that the equation of the line that passes through the points and is given by the relation;
and the line passing through the points,
And any point on the line is of the form .
So, the equation of ZX plane is
Since the line passes through YZ- plane,
we have then,
or and
So, therefore the required point is
Answer:
We know that the equation of the line that passes through the points and is given by the relation;
and the line passing through the points, .
And any point on the line is of the form.
This point lies on the plane,
or .
Hence, the coordinates of the required point are or .
Answer:
Given
two planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
the normal vectors of these plane are
Since the normal vector of the required plane is perpendicular to the normal vector of given planes, the required plane's normal vector will be :
Now, as we know
the equation of a plane in vector form is :
Now Since this plane passes through the point (-1,3,2)
Hence the equation of the plane is
Question:14 If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane then find the value of p.
Answer:
Given that the points and are equidistant from the plane
So we can write the position vector through the point is
Similarly, the position vector through the point is
The equation of the given plane is
and We know that the perpendicular distance between a point whose position vector is and the plane, and
Therefore, the distance between the point and the given plane is
nbsp; .........................(1)
Similarly, the distance between the point , and the given plane is
.........................(2)
And it is given that the distance between the required plane and the points, and is equal.
therefore we have,
or or
Answer:
So, the given planes are:
and
The equation of any plane passing through the line of intersection of these planes is
..............(1)
Its direction ratios are and = 0
The required plane is parallel to the x-axis.
Therefore, its normal is perpendicular to the x-axis.
The direction ratios of the x-axis are 1,0, and 0.
Substituting in equation (1), we obtain
So, the Cartesian equation is
Answer:
We have the coordinates of the points and respectively.
Therefore, the direction ratios of OP are
And we know that the equation of the plane passing through the point is
where a,b,c are the direction ratios of normal.
Here, the direction ratios of normal are and and the point P is .
Thus, the equation of the required plane is
Answer:
The equation of the plane passing through the line of intersection of the given plane in
,,,,,,,,,,,,,(1)
The plane in equation (1) is perpendicular to the plane, Therefore
Substituting in equation (1), we obtain
.......................(4)
So, this is the vector equation of the required plane.
The Cartesian equation of this plane can be obtained by substituting in equation (1).
Therefore we get the answer
Question:18 Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the line and the plane .
Answer:
Given,
Equation of a line :
Equation of the plane
Let's first find out the point of intersection of line and plane.
putting the value of into the equation of a plane from the equation from line
Now, from the equation, any point p in line is
So the point of intersection is
SO, Now,
The distance between the points (-1,-5,-10) and (2,-1,2) is
Hence the required distance is 13.
Question:19 Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .
Answer:
Given
A point through which line passes
two plane
And
it can be seen that normals of the planes are
since the line is parallel to both planes, its parallel vector will be perpendicular to normals of both planes.
So, a vector perpendicular to both these normal vector is
Now a line which passes through and parallels to is
So the required line is
Questio n: 20 Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:
Answer:
Given
Two straight lines in 3D whose direction cosines (3,-16,7) and (3,8,-5)
Now the two vectors which are parallel to the two lines are
and
As we know, a vector perpendicular to both vectors and is , so
A vector parallel to this vector is
Now as we know the vector equation of the line which passes through point p and parallel to vector d is
Here in our question, give point p = (1,2,-4) which means position vector of this point is
So, the required line is
Question:21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then .
Answer:
The equation of plane having a, b and c intercepts with x, y and z-axis respectively is given by
The distance p of the plane from the origin is given by
Hence proved
Question:22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units (B) 4 units (C) 8 units (D)
Answer:
Given equations are
and
Now, it is clear from equation (i) and (ii) that given planes are parallel
We know that the distance between two parallel planes is given by
Put the values in this equation
we will get,
Therefore, the correct answer is (D)
Question:23 The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are
(A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through
Answer:
Given equations of planes are
and
Now, from equation (i) and (ii) it is clear that given planes are parallel to each other
Therefore, the correct answer is (B)
If you are looking for exercises solutions for chapter 3d Geometry class 12 then they are listed below.
All the important topics are covered in the maths chapter 11 class 12 NCERT solutions.
A total of 36 questions in 3 exercises are given in this Maths chapter 11 class 12 solutions.
All these NCERT questions are solved and explained in the NCERT solutions for class 12 maths chapter 11 three dimensional geometry article to clear your doubts.
In this NCERT Solutions for Class 12 Maths Chapter 11 PDF Download, we deal with formulas like-
If l, m, n are the direction cosines of a line, then.
andDirection cosines of a line joining two pointsare, where
If l, m, n are the direction cosines and a, b, c are the direction of a line then-
Also read,
11.1 Introduction
11.2 Direction Cosines and Direction Ratios of a Line
11.2.1 Relation between the direction cosines of a line
11.2.2 Direction cosines of a line passing through two points
11.3 Equation of a Line in Space
11.3.1Equation of a line through a given point and parallel to a given vector b
11.3.2 Equation of a line passing through two given points
11.4 Angle between Two Lines
11.5 Shortest Distance between Two Lines
11.5.1 Distance between two skew lines
11.5.2 Distance between parallel lines
11.6 Plane
11.6.1 Equation of a plane in normal form
11.6.2 Equation of a plane perpendicular to a given vector and passing through a given point
11.6.3 Equation of a plane passing through three noncollinear points
11.6.4 Intercept form of the equation of a plane
11.6.5 Plane passing through the intersection of two given planes
11.7 Coplanarity of Two Lines
11.8 Angle between Two Planes
11.9 Distance of a Point from a Plane
11.10 Angle between a Line and a Plane
Chapter 1 | NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions |
Chapter 2 | NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions |
Chapter 3 | |
Chapter 4 | |
Chapter 5 | NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability |
Chapter 6 | NCERT solutions for class 12 maths chapter 6 Application of Derivatives |
Chapter 7 | |
Chapter 8 | NCERT solutions for class 12 maths chapter 8 Application of Integrals |
Chapter 9 | NCERT solutions for class 12 maths chapter 9 Differential Equations |
Chapter 10 | NCERT solutions for class 12 maths chapter 10 Vector Algebra |
Chapter 11 | NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry |
Chapter 12 | NCERT solutions for class 12 maths chapter 12 Linear Programming |
Chapter 13 |
Chapter 11 of Class 12 Maths is titled "Three Dimensional Geometry." NCERT Solutions for this chapter provide step-by-step solutions to all the exercises and problems included in the textbook. Some of the key features of NCERT Solutions for Class 12 Maths Chapter 11 are:
Comprehensive Coverage: The ch 11 maths class 12 solutions cover all the important topics and concepts discussed in the chapter, including the coordinate axes and coordinate planes in three dimensions, distance between two points, section formula, direction cosines and direction ratios of a line, angle between two lines, equation of a line and a plane, and the distance of a point from a plane.
Simple and Clear Explanation: The class 12 maths ch 11 question answer are written in a simple and clear language that makes it easy for students to understand even the most complex concepts.
Step-by-step Approach: The class 12 maths ch 11 question answer are provided in a step-by-step manner, making it easy for students to follow and learn.
Class 12 Maths Chapter 11 NCERT solutions are very easy for you to understand the concepts as they are explained in a step-by-step manner.
NCERT Class 12 Maths solutions chapter 11 will give you some new insight into the concepts.
Scoring good marks in the exam is now a reality with the help of these solutions of NCERT for class 12 maths chapter 11 three dimensional geometry as these questions are answered by the experts who know how best to answer the questions in the board exam.
You should solve the miscellaneous exercise also, to develop a grip on the concepts. Here, you will get solutions for miscellaneous exercise too.
Three dimensional geometry class 12 ncert solutions PDF Download will also be made available soon.
The concepts of vector algebra and Three Dimensional Geometry can be used interchangeably. if two chapters are combined, vector algebra & three-dimensional geometry has a 17% weightage in the 12th board maths final exam. after getting command of these concepts it becomes easy for students to score well in the exam therefore NCERT Notes, NCERT syllabus, and NCERT textbooks are recommended.
Undoubtedly, NCERT Solutions for Class 12 Maths Chapter 11 stand out as the top study material aiding students in effortlessly revising complex concepts. The solutions present a well-reasoned explanation to facilitate student learning. The team of experts at Careers360 has carefully crafted step-by-step solutions that encourage students to employ an analytical thinking approach. Moreover, these solutions can be cross-referenced to gain insights into alternative methods to solve textbook problems.
Only knowing the answer is not enough to score good marks in the exam. One should know how best to answer in order to get good marks. NCERT solutions are provided by the experts who know how best to write answer in the board exam in order to get good marks.
Class 12 chapter 11 maths Maths covers the following important topics:
Introduction (11.1)
Direction cosines and direction ratios of a line (11.2)
Equation of a line in space (11.3)
Angle between two lines (11.4)
Shortest distance between two lines (11.5)
Plane (11.6)
Coplanarity of two lines (11.7)
Angle between two planes (11.8)
Distance of a point from a plane (11.9)
Angle between a line and a plane (11.10)
The ncert solutions class 12 maths chapter 11 miscellaneous is useful for students in several ways. These exercises include additional problems that are not part of the main textbook exercises but are relevant to the chapter's concepts.
Firstly, NCERT solutions for class 12 maths chapter 11 miscellaneous exercise help students to gain a deeper understanding of the concepts covered in Chapter 11.
Secondly, these ch 11 miscellaneous class 12 exercises help students to test their problem-solving abilities and improve their skills.
Thirdly, solving the miscellaneous exercise problems for 3d class 12 ncert solutions can help students to prepare for their exams.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.
The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.
How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.
Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities.
Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters
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