NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra: In our day to day life, we come across many queries such as 'What is your height?' How should a football player hit the ball to give a pass to another player of his team? The answer to the second question is a force. It has magnitude and direction. Such quantities are called as vector quantities. In this article, you will get NCERT solutions for class 12 maths chapter 10 vector algebra. Important topics that are going to be discussed in this chapter are vector quantities, operations on vectors, geometric properties and algebraic properties like addition, multiplication, etc. In CBSE NCERT solutions for class 12 maths chapter 10 vector algebra article, questions from all these topics are covered. Vector algebra plays an important role in physics also. Solutions of NCERT for class 12 maths chapter 10 vector algebra will build your concepts of vectors which will help you to solve the many problems of physics in competitive exams like JEE mains, Jee Advanced, BITSAT, VITEEE. Check all NCERT solutions from class 6 to 12 at a single place, which will help you to get a better understanding of concepts in a much easy way.
Two chapters vector algebra & threedimensional geometry has 17% weightage in 12th board maths final examination. The CBSE NCERT solutions for class 12 maths chapter 10 vector algebra will help you to score good marks in the exam. The concepts of vector algebra are very useful in both maths and physics. In chapter 10 vector algebra total of 54 questions in 4 exercises and 30 solved examples are given in the NCERT textbook. These NCERT questions are prepared and explained in a detailed manner to help students in their board exam as well as in competitive exams. Before discussing this chapter in detail, you need to understand what is a vector quantity and what is a scalar quantity.
Vector Quantity Quantity which involves both the value magnitude and direction. Vector quantities like weight, velocity, acceleration, displacement, force, momentum, etc.
Scalar Quantity Quantity which involves only one value (magnitude) which is a real number. Scalar quantities like distance, length, time, mass, speed, area, temperature, work, money, volume, voltage, density, resistance, etc.
The focus of class 12 maths chapter 10 vector algebra is on vector quantities. To develop a grip on the topic, students must try to solve the miscellaneous exercise also. In the NCERT solutions for class 12 maths chapter 10 vector algebra, you will find solutions to miscellaneous exercises too.
10.1 Introduction
10.2 Some Basic Concepts
10.3 Types of Vectors
10.4 Addition of Vectors
10.5 Multiplication of a Vector by a Scalar
10.5.1 Components of a vector
10.5.2 Vector joining two points
10.5.3 Section formula
10.6 Product of Two Vectors
10.6.1 Scalar (or dot) product of two vectors
10.6.2 Projection of a vector on a line
10.6.3 Vector (or cross) product of two vectors
Question:1 Represent graphically a displacement of 40 km, east of north.
Answer:
Represent graphically a displacement of 40 km, east of north.
N,S,E,W are all 4 direction north,south,east,west respectively.
is displacement vector which
= 40 km.
makes an angle of 30 degree east of north as shown in figure.
Question:2 (1) Classify the following measures as scalars and vectors.
Answer:
10kg is a scalar quantity as it has only magnitude.
Question:2 (2) Classify the following measures as scalars and vectors. 2 meters north west
Answer:
This is a vector quantity as it has both magnitude and direction.
Question:2 (3) Classify the following measures as scalars and vectors.
Answer:
This is a scalar quantity as it has only magnitude.
Question:2 (4) Classify the following measures as scalars and vectors. 40 watt
Answer:
This is a scalar quantity as it has only magnitude.
Question:2 (5) Classify the following measures as scalars and vectors.
Answer:
This is a scalar quantity as it has only magnitude.
Question:2 (6) Classify the following measures as scalars and vectors.
Answer:
This is a Vector quantity as it has magnitude as well as direction.by looking at the unit, we conclude that measure is acceleration which is a vector.
Question:3 Classify the following as scalar and vector quantities.
(1) time period
Answer:
This is a scalar quantity as it has only magnitude.
Question:3 Classify the following as scalar and vector quantities.
Answer:
Distance is a scalar quantity as it has only magnitude.
Question:3 Classify the following as scalar and vector quantities.
Answer:
Force is a vector quantity as it has both magnitude as well as direction.
Question:3 Classify the following as scalar and vector quantities.
(4) velocity
Answer:
Velocity is a vector quantity as it has both magnitude and direction.
Question:3 Classify the following as scalar and vector quantities.
Answer:
work done is a scalar quantity, as it is the product of two vectors.
Question:4 In Fig 10.6 (a square), identify the following vectors.
(1) Coinitial
Answer:
Since vector and vector are starting from the same point, they are coinitial.
Question:4 In Fig 10.6 (a square), identify the following vectors.
(2) Equal
Answer:
Since Vector and Vector both have the same magnitude and same direction, they are equal.
Question:4 In Fig 10.6 (a square), identify the following vectors.
Answer:
Since vector and vector have the same magnitude but different direction, they are colinear and not equal.
Question:5 Answer the following as true or false.
(1) and are collinear.
Answer:
True, and are collinear. they both are parallel to one line hence they are colinear.
Question:5 Answer the following as true or false.
(2) Two collinear vectors are always equal in magnitude.
Answer:
False, because colinear means they are parallel to the same line but their magnitude can be anything and hence this is a false statement.
Question:5 Answer the following as true or false.
(3) Two vectors having same magnitude are collinear.
Answer:
False, because any two noncolinear vectors can have the same magnitude.
Question:5 Answer the following as true or false.
(4) Two collinear vectors having the same magnitude are equal.
Answer:
False, because two colinear vectors with the same magnitude can have opposite direction
Question:1 Compute the magnitude of the following vectors:
Answer:
Here
Magnitude of
Question:2 Write two different vectors having same magnitude
Answer:
Two different Vectors having the same magnitude are
The magnitude of both vector
Question:3 Write two different vectors having same direction.
Answer:
Two different vectors having the same direction are:
Question:4 Find the values of x and y so that the vectors and are equal.
Answer:
will be equal to when their corresponding components are equal.
Hence when,
and
Answer:
Let point P = (2, 1) and Q = (– 5, 7).
Now,
Hence scalar components are (7,6) and the vector is
Question:7 Find the unit vector in the direction of the vector
Answer:
Given
Magnitude of
A unit vector in the direction of
Answer:
Given P = (1, 2, 3) and Q = (4, 5, 6)
A vector in direction of PQ
Magnitude of PQ
Now, unit vector in direction of PQ
Question:9 For given vectors, and , find the unit vector in the direction of the vector .
Answer:
Given
Now,
Now a unit vector in the direction of
Question:10 Find a vector in the direction of vector which has magnitude 8 units.
Answer:
Given a vector
the unit vector in the direction of
A vector in direction of and whose magnitude is 8 =
Question:11 Show that the vectors and are collinear.
Answer:
Let
It can be seen that
Hence here
As we know
Whenever we have , the vector and will be colinear.
Here
Hence vectors and are collinear.
Answer:
Given
point A=(1, 2, –3)
point B=(–1, –2, 1)
Vector joining A and B Directed from A to B
Hence Direction cosines of vector AB are
Question:14 Show that the vector is equally inclined to the axes OX, OY and OZ.
Answer:
Let
Hence direction cosines of this vectors is
Let , and be the angle made by xaxis, yaxis and z axis respectively
Now as we know,
,
Hence Given vector is equally inclined to axis OX,OY and OZ.
Answer:
As we know
The position vector of the point R which divides the line segment PQ in ratio m:n internally:
Here
position vector os P = =
the position vector of Q =
m:n = 2:1
And Hence
Answer:
As we know
The position vector of the point R which divides the line segment PQ in ratio m:n externally:
Here
position vector os P = =
the position vector of Q =
m:n = 2:1
And Hence
Question:16 Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).
Answer:
Given
The position vector of point P =
Position Vector of point Q =
The position vector of R which divides PQ in half is given by:
Answer:
Given
the position vector of A, B, and C are
Now,
AS we can see
Hence ABC is a right angle triangle.
Question:18 In triangle ABC (Fig 10.18), which of the following is not true:
Answer:
From triangles law of addition we have,
From here
also
Also
Hence options A,B and D are true SO,
Option C is False.
Answer:
If two vectors are collinear then, they have same direction or are parallel or antiparallel.
Therefore,
They can be expressed in the form where a and b are vectors and is some scalar quantity.
Therefore, (a) is true.
Now,
(b) is a scalar quantity so its value may be equal to
Therefore,
(b) is also true.
C) The vectors and are proportional,
Therefore, (c) is not true.
D) The vectors and can have different magnitude as well as different directions.
Therefore, (d) is not true.
Therefore, the correct options are (C) and (D).
Answer:
Given
As we know
where is the angle between two vectors
So,
Hence the angle between the vectors is .
Question:2 Find the angle between the vectors
Answer:
Given two vectors
Now As we know,
The angle between two vectors and is given by
Hence the angle between
Question:3 Find the projection of the vector on the vector
Answer:
Let
Projection of vector on
Hence, Projection of vector on is 0.
Question:4 Find the projection of the vector on the vector
Answer:
Let
The projection of on is
Hence, projection of vector on is
Answer:
Given
Now magnitude of
Hence, they all are unit vectors.
Now,
Hence all three are mutually perpendicular to each other.
Answer:
Given two vectors
Now Angle between
Now As we know that
Hence, the magnitude of two vectors
Question:9 Find , if for a unit vector
Answer:
Given in the question that
And we need to find
So the value of is
Question:10 If are such that is perpendicular to , then find the value of
Answer:
Given in the question is
and is perpendicular to
and we need to find the value of ,
so the value of 
As is perpendicular to
the value of ,
Question:11 Show that is perpendicular to , for any two nonzero vectors .
Answer:
Given in the question that 
are two nonzero vectors
According to the question
Hence is perpendicular to .
Question:12 If , then what can be concluded about the vector ?
Answer:
Given in the question
Therefore is a zero vector. Hence any vector will satisfy
Question:13 If are unit vectors such that , find the value of
Answer:
Given in the question
are unit vectors
and
and we need to find the value of
Answer the value of is
Question:14 If either vector . But the converse need not be true. Justify your answer with an example
Answer:
Let
we see that
we now observe that
Hence here converse of the given statement is not true.
Answer:
Given points,
A=(1, 2, 3),
B=(–1, 0, 0),
C=(0, 1, 2),
As need to find Angle between
Hence angle between them ;
Answer Angle between the vectors is
Question:16 Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.
Answer:
Given in the question
A=(1, 2, 7), B=(2, 6, 3) and C(3, 10, –1)
To show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear
As we see that
Hence point A, B , and C are colinear.
Question:17 Show that the vectors form the vertices of a right angled triangle.
Answer:
Given the position vector of A, B , and C are
To show that the vectors form the vertices of a right angled triangle
Here we see that
Hence A,B, and C are the vertices of a right angle triangle.
Question:18 If is a nonzero vector of magnitude ‘a’ and a nonzero scalar, then is unit vector if
Answer:
Given is a nonzero vector of magnitude ‘a’ and a nonzero scalar
is a unit vector when
Hence the correct option is D.
Question:1 Find
Answer:
Given in the question,
and we need to find
Now,
So the value of is
Question:2 Find a unit vector perpendicular to each of the vector , where
Answer:
Given in the question
Now , A vector which perpendicular to both is
And a unit vector in this direction :
Hence Unit vector perpendicular to each of the vector is .
Answer:
Given in the question,
angle between and :
angle between and
angle with and :
Now, As we know,
Now components of are:
Question:4 Show that
Answer:
To show that
LHS=
As product of a vector with itself is always Zero,
As cross product of a and b is equal to negative of cross product of b and a.
= RHS
LHS is equal to RHS, Hence Proved.
Question:5 Find and if
Answer:
Given in the question
and we need to find values of and
From Here we get,
From here, the value of and is
Question:6 Given that and . What can you conclude about the vectors ?
Answer:
Given in the question
and
When , either are perpendicular to each other
When either are parallel to each other
Since two vectors can never be both parallel and perpendicular at same time,we conclude that
Question:7 Let the vectors be given as Then show that
Answer:
Given in the question
We need to show that
Now,
Now
Hence they are equal.
Question:8 If either then . Is the converse true? Justify your answer with an example.
Answer:
No, the converse of the statement is not true, as there can be two non zero vectors, the cross product of whose are zero. they are colinear vectors.
Consider an example
Here
Hence converse of the given statement is not true.
Question:9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
Answer:
Given in the question
vertices A=(1, 1, 2), B=(2, 3, 5) and C=(1, 5, 5). and we need to find the area of the triangle
Now as we know
Area of triangle
The area of the triangle is square units
Question:10 Find the area of the parallelogram whose adjacent sides are determined by the vectors and .
Answer:
Given in the question
Area of parallelogram with adjescent side and ,
The area of the parallelogram whose adjacent sides are determined by the vectors and is
Question:11 Let the vectors be such that , then is a unit vector, if the angle between is
Answer:
Given in the question,
As given is a unit vector, which means,
Hence the angle between two vectors is . Correct option is B.
Question:12 Area of a rectangle having vertices A, B, C and D with position vectors
Answer:
Given 4 vertices of rectangle are
Now,
Area of the Rectangle
Hence option C is correct.
Question:1 Write down a unit vector in XYplane, making an angle of with the positive direction of xaxis.
Answer:
As we know
a unit vector in XYPlane making an angle with xaxis :
Hence for
Answer the unit vector in XYplane, making an angle of with the positive direction of xaxis is
Question:2 Find the scalar components and magnitude of the vector joining the points
Answer:
Given in the question
And we need to finrd the scalar components and magnitude of the vector joining the points P and Q
Magnitiude of vector PQ
Scalar components are
Answer:
As the girl walks 4km towards west
Position vector =
Now as she moves 3km in direction 30 degree east of north.
hence final position vector is;
Question:4 If , then is it true that ? Justify your answer.
Answer:
No, if then we can not conclude that .
the condition satisfies in the triangle.
also, in a triangle,
Since, the condition is contradicting with the triangle inequality, if then we can not conclude that
Question:5 Find the value of x for which is a unit vector.
Answer:
Given in the question,
a unit vector,
We need to find the value of x
The value of x is
Question:6 Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
Answer:
Given two vectors
Resultant of and :
Now, a unit vector in the direction of
Now, a unit vector of magnitude in direction of
Hence the required vector is
Question:7 If , find a unit vector parallel to the vector .
Answer:
Given in the question,
Now,
let vector
Now, a unit vector in direction of
Now,
A unit vector parallel to
OR
Answer:
Given in the question,
points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)
now let's calculate the magnitude of the vectors
As we see that AB = BC + AC, we conclude that three points are colinear.
we can also see from here,
Point B divides AC in the ratio 2 : 3.
Answer:
Given, two vectors
the point R which divides line segment PQ in ratio 1:2 is given by
Hence position vector of R is .
Now, Position vector of the midpoint of RQ
which is the position vector of Point P . Hence, P is the midpoint of RQ
Answer:
Given, two adjacent sides of the parallelogram
The diagonal will be the resultant of these two vectors. so
resultant R:
Now unit vector in direction of R
Hence unit vector along the diagonal of the parallelogram
Now,
Area of parallelogram
Hence the area of the parallelogram is .
Question:11 Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
Answer:
Let a vector is equally inclined to axis OX, OY and OZ.
let direction cosines of this vector be
Now
Hence direction cosines are:
Question:12 Let . Find a vector which is perpendicular to both
Answer:
Given,
Let
now, since it is given that d is perpendicular to and , we got the condition,
and
And
And
here we got 2 equation and 3 variable. one more equation will come from the condition:
so now we have three equation and three variable,
On solving this three equation we get,
,
Hence Required vector :
Answer:
Let, the sum of vectors and be
unit vector along
Now, the scalar product of this with
squaring both the side,
Question:14 If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to .
Answer:
Given
and
Now, let vector is inclined to at respectively.
Now,
Now, Since,
Hence vector is equally inclined to .
Question:15 Prove that , if and only if are perpendicular, given
Answer:
Given in the question,
are perpendicular and we need to prove that
LHS=
if are perpendicular,
= RHS
LHS ie equal to RHS
Hence proved.
Question:16 Choose the correct answer If is the angle between two vectors , then only when
Answer:
Given in the question
is the angle between two vectors
this will satisfy when
Hence option B is the correct answer.
Question:19 Choose the correct. If is the angle between any two vectors , then when
is equal to
Answer:
Given in the question
is the angle between any two vectors and
To find the value of
Hence option D is correct.
chapter 1 
Solutions of NCERT for class 12 maths chapter 1 Relations and Functions 
chapter 2 
CBSE NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions 
chapter 3 

chapter 4 
Solutions of NCERT for class 12 maths chapter 4 Determinants 
chapter 5 
CBSE 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 
CBSE 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 
Solutions of NCERT for class 12 maths chapter 10 Vector Algebra 
chapter 11 
CBSE 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 
Solutions of NCERT for class 12 maths chapter 13 Probability 
NCERT solutions are explained in a stepbystep manner, so it will be very easy for you to understand the concepts.
NCERT Solutions for class 12 maths chapter 10 vector algebra will give you some new way to solve the problem.
Performance in the 12th board exam plays a very important role in deciding the future, so you can get admission to a good college. Scoring good marks in the exam is now a reality with the help of these solutions of NCERT for class 12 maths chapter 10 vector algebra.
To develop a grip on the concept, you should solve the miscellaneous exercise also. In CBSE NCERT solutions for class 12 maths chapter 10 vector algebra article, you will get a solution of miscellaneous exercise also.
Happy learning !!!