If the image of the point P(1, −2, 3) in the plane, 2x+3y−4z+22=0 measured parallel to the line, is Q, then PQ is equal to :
Option: 1
Option: 5
Option: 9
Option: 13
Intersection of line and plane -
Let the line
plane
intersect at P
to find P assume general point on line as
now put it in plane to find ,
-
Distance formula -
The distance between two points is =
- wherein
= -
PQ=2PM
Now, for finding M
Putting in equation of plane,
So,
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The line of intersection of the planes and , is:
Option: 1
Option: 2
Option: 3
Option: 4
As we have learned
Equation of line as intersection of two planes -
Let the two intersecting planes be
and
then the parallel vector of line formed their intersection can be obtained by
and points can be obtained by putting and solving
and
say
Now the equation will be
-
putting z = 0
3x-y = 1 and x + 4y = 2
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The shortest distance between the lines and is :
Option: 1
Option: 2
Option: 3
Option: 4
Shortest Distance between Two Lines -
Distance between two skew lines
If L1 and L2 are two skew lines, then there is one and only one line perpendicular to each of lines L1 and L2 which is known as the line of shortest distance.
Vector form
If is the shortest distance vector between L1 and L2, then it being perpendicular to both and , therefore, the unit vector along would be
where "d" is the magnitude of the shortest distance vector. Let θ be the angle between and .
Then
Hence, the required shortest distance is
-
=
Correct Option (4)
View Full Answer(1)Let be a plane passing through the points and and be any point Then the image of in the plane is :
Option: 1
Option: 2
Option: 3
Option: 4
Equation of a plane passing through three non collinear point -
Cartesian Form
Let (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) be the coordinates of points A, B and C respectively.
Let P(x, y z) be any point on the plane.
Then, the vectors and are coplanar.
-
Image of a Point in the Plane -
The image of the point P(x1, y1, z1) in the plane ax + by + cz + d = 0 is Q(x3, y3, z3) then coordinates of point Q is given by
-
Points of plane P ( 2,1,0), (4,1,1) and ( 5,0,1) and point R ( 2,1,6) .
Then the image of R in the plane P is:
Correct option (2)
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A vector lies in the plane of the vectors, and If bisects the angle between and , then:
Option: 1
Option: 2
Option: 3
Option: 4
Equation of The Plane Bisecting the Angle Between Two Planes -
-
angle bisector can be
Compare it with
Compare it with
Correct Option (4)
View Full Answer(1)The mirror image of the point (1,2,3) in a plane is . which of the following points lies on this plane ?
Option: 1
Option: 2
Option: 3
Option: 4
Image of a Point in the Plane -
The image of the point P(x1, y1, z1) in the plane ax + by + cz + d = 0 is Q(x3, y3, z3) then coordinates of point Q is given by
Proof:
Let point Q(x3, y3, z3) is the image of the point P(x1, y1, z1) in the plane π : ax + by + cz + d = 0.
Let line PQ meets the plane ax+by+cz+d=0 at point R, direction ratio of normal to plane π are (a, b, c), since, PQ perpendicular to plane π.
So direction ratio of PQ are a, b, c
Since, point R lies in the plane π, we get
In the similar method we can also find the coordinates of point R.
-
P(1,2,3) and Q
Direction ratio of normal to the plane
we get
and Midpoint of PQ is
equation of plane x + y + z = 1
Correct Option (1)
View Full Answer(1)If the foot of the perpendicular drawn from the point (1,0,3) on line passing through is then is equal to _______.
Option: 1 2
Option: 2 4
Option: 3 8
Option: 4 10
Angle Between Two Lines -
Vector Form
Condition for Perpendicularity
The lines are perpendicular then cos? = 90o
Condition for parallelism
The lines are parallel then for some scalar λ.
-
Since PQ is perpendicular to L, therefore
View Full Answer(1)
The angle bisectors are
OR
Take a point (-1,0,0) on plane 1 and find its distance from both angle bisectors
Clearly
So plane in equation (1) is acute angle Bisector
Out of the given option, satisfies this plane
Let PQ be perpendicular to the given line
Now Q lies on both the planes,
So and
Solving (i),(ii) & (iii)
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Three dimensional Coordinate Geometry
Introduction to Three Dimensional Geometry
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Three Dimensional Geometry
If the image of the point P(1, −2, 3) in the plane, 2x+3y−4z+22=0 measured parallel to the line,
The line of intersection of the planes
The shortest distance between the lines and
Let be a plane passing through the points and
The mirror image of the point (1,2,3) in a plane is
If the foot of the perpendicular drawn from the point (1,0,3) on line passing through is
Let the foot of perpendicular from a point to the straight line