The NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra are given here where the students learn about the difference between a scalar and a vector quantity, their properties, operations of vectors etc. The topic has an important role in helping the students score high marks not only in the board exams but also the competitive exams.
It is important to be prepared for the various problems asked during the 12th board examination. Solving through different exercises and problem sets give students the confidence to write the exams better. These class 12 NCERT solutions for Vector Algebra are very easy to understand. Students can also avail these NCERT solutions and download it for free to practice them offline as well.
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NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra
Exercise 10.1 Page No: 428
1. Represent graphically a displacement of 40 km, 30° east of north.
Solution:
The vector
represents the displacement of 40 km, 30o east of north.
2. Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres north-west (iii) 40°
(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2
Solution:
(i) 10 kg is a scalar quantity because it has only magnitude.
(ii) 2 meters north-west is a vector quantity as it has both magnitude and direction.
(iii) 40° is a scalar quantity as it has only magnitude.
(iv) 40 watts is a scalar quantity as it has only magnitude.
(v) 10–19 coulomb is a scalar quantity as it has only magnitude.
(vi) 20 m/s2Â is a vector quantity as it has both magnitude and direction.
3. Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force
(iv) velocity (v) work done
Solution:
(i) Time period is a scalar quantity as it has only magnitude.
(ii) Distance is a scalar quantity as it has only magnitude.
(iii) Force is a vector quantity as it has both magnitude and direction.
(iv) Velocity is a vector quantity as it has both magnitude as well as direction.
(v) Work done is a scalar quantity as it has only magnitude.
4. In Figure, identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
Solution:
directions are not the same.
5. Answer the following as true or false.
(i)Â 
 and
are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Solution:
(i) True.
Vectors 
 and
are parallel to the same line.
(ii) False.
Collinear vectors are those vectors that are parallel to the same line.
(iii) False.
Two vectors having the same magnitude need not necessarily be parallel to the same line.
(iv) False.
Only if the magnitude and direction of two vectors are the same, regardless of the positions of their initial points the two vector are said to be equal.
Exercise 10.2 Page No: 440
1. Compute the magnitude of the following vectors:
Solution:
Given vectors are:
2. Write two different vectors having same magnitude.
Solution:
3. Write two different vectors having same direction.
Solution:
4. Find the values of x and y so that the vectors 
are equal
Solution:
Given vectors 
will be equal only if their corresponding components are equal.
Thus, the required values of x and y are 2 and 3 respectively.
5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).
Solution:
The vector with initial point P (2, 1) and terminal point Q (–5, 7) can be shown as,
Thus, the required scalar components are –7 and 6 while the vector components areÂ
6. Find the sum of the vectors
Solution:
7. Find the unit vector in the direction of the vector
Solution:
8. Find the unit vector in the direction of vector , where P and Q are the points
(1, 2, 3) and (4, 5, 6), respectively
Solution:
9. For given vectors, 
and 
, find the unit vector in the direction of the vector 
Solution:
10. Find a vector in the direction of vector 
which has magnitude 8 units.
Solution:
11. Show that the vectors
are collinear.
Solution:
Therefore, the given vectors are collinear.
12. Find the direction cosines of the vector 
Solution:
13. Find the direction cosines of the vector joining the points A (1, 2, –3) and
B (–1, –2, 1) directed from A to B.
Solution:
Given points are A (1, 2, –3) and B (–1, –2, 1).
Now,
14. Show that the vector 
is equally inclined to the axes OX, OY, and OZ.
Solution:
15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are 
 respectively, in the ration 2:1
(i) internally
(ii) externally
Solution:
The position vector of point R dividing the line segment joining two points
P and Q in the ratio m: n is given by:
16. Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
Solution:
The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,
17. Show that the points A, B and C with position vectors,
, 
respectively form the vertices of a right angled triangle.
Solution:
Given position vectors of points A, B, and C are:
18. In triangle ABC (Fig 10.18) which of the following is not true:
Solution:
19. If 
are two collinear vectors, then which of the following are incorrect:
A. 
, for some scalar λ
B. 
C. the respective components of 
are proportional
D. both the vectors 
have same direction, but different magnitudes
Solution:
Exercise 10.3 Page No: 447
1. Find the angle between two vectors
and
with magnitudes √3 and 2, respectively having
.
Solution:
2. Find the angle between the vectors
Solution:
3. Find the projection of the vector
on the vector
.
Solution:
4. Find the projection of the vector
on the vector
.
Solution:
5. Show that each of the given three vectors is a unit vector:
Also, show that they are mutually perpendicular to each other.
Solution:
6. Find
Solution:
7. Evaluate the product
Solution:
8. Find the magnitude of two vectors
, having the same magnitude and such that the angle between them is 60° and their scalar product is ½.
Solution:
9. Find
, if for a unit vector
Solution:
10. If
are such that
is perpendicular to
, then find the value of λ.
Solution:
11. Show that 
is perpendicular to
, for any two nonzero vectors
.
Solution:
12. If
, then what can be concluded about the vector
?
Solution:
13. If 
are unit vectors such that 
, find the value of 
.
Solution:
14. If either vector
, then
. But the converse need not be true. Justify your answer with an example.
Solution:
15. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectors
and
]
Solution:
16. Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.
Solution:
Given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).
Now,
Therefore, the given points A, B, and C are collinear.
17. Show that the vectors
form the vertices of a right angled triangle.
Solution:
18. If
is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λ
is unit vector if
Solution:
Exercise 10.4 Page No: 454
1. Find
, if 
and
Solution:
We have,
2. Find a unit vector perpendicular to each of the vector 
and
, where 
and
.
Solution:
We have,
3. If a unit vector 
 makes an angles
with 
with 
and an acute angle θ with
, then find θ and hence, the compounds of
.
Solution:
4. Show that
Solution:
Taking the LHS, we have
5. Find λ and μ if 
.
Solution:
6. Given that 
 and
. What can you conclude about the vectors
?
Solution:
Given,
7. Let the vectors 
given as
 
. Then show that 
Solution:
We have,
8. If either 
 or
, then
. Is the converse true? Justify your answer with an example.
Solution:
9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Solution:
10. Find the area of the parallelogram whose adjacent sides are determined by the vector 
.
Solution:
11. Let the vectors 
and 
be such that 
and
, then
 is a unit vector, if the angle between 
and 
is
Solution:
12. Area of a rectangle having vertices A, B, C, and D with position vectors 
and 
 respectively is
Solution:
Miscellaneous Exercise Page No: 458
1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
Solution:
2. Find the scalar components and magnitude of the vector joining the points P (x1, y1, z1) and Q (x2, y2, z2).
Solution:
3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.
Solution:
Let O and B be the initial and final positions of the girl respectively.
Then, the girl’s position can be shown as:
4. If
, then is it true that
? Justify your answer.
Solution:
5. Find the value of x for which
is a unit vector.
Solution:
6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
.
Solution:
7. If
, find a unit vector parallel to the vector
.
Solution:
8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Solution:
9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.
Solution:
10. The two adjacent sides of a parallelogram are
and 
.
Find the unit vector parallel to its diagonal. Also, find its area.
Solution:
11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
.
Solution:
Let’s assume a vector to be equally inclined to axes OX, OY, and OZ at angle α.
Then, the direction cosines of the vector are cos α, cos α, and cos α.
Now, we know that
Therefore, the direction cosines of the vector which are equally inclined to the axes are
.
12. Let 
and
. Find a vector 
which is perpendicular to both 
and
, and
.
Solution:
13. The scalar product of the vector
with a unit vector along the sum of vectors 
and 
is equal to one. Find the value of λ.
Solution:
14. If 
are mutually perpendicular vectors of equal magnitudes, show that the vector 
is equally inclined to 
and
.
Solution:
15. Prove that
, if and only if 
are perpendicular, given
.
Solution:
16. If θ is the angle between two vectors 
and 
, then 
only when
Solution:
17. Let 
 and 
 be two unit vectors andθ is the angle between them. Then 
is a unit vector if
Solution:
18. The value of 
is
(A) 0 (B) –1 (C) 1 (D) 3
Solution:
Given,
The correct answer is C.
19. If θ is the angle between any two vectors 
and
, then 
when θ is equal to
Solution:
The major concepts of Maths covered in Chapter 10- Vector Algebra of NCERT Solutions for Class 12 includes:
10.1 Introduction
10.2 Basic Concepts
-
- Position Vector
- Direction Cosines
10.3 Types of Vectors
-
- Zero Vector
- Unit Vector
- Coinitial Vectors
- Collinear Vectors
- Equal Vectors
- Negative of a Vector
10.4 Addition of Vectors
-
- Properties of vector addition
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
Exercise 10.1 Solutions 5 Questions
Exercise 10.2 Solutions 19 Questions
Exercise 10.3 Solutions 18 Questions
Exercise 10.4 Solutions 12 Questions
Miscellaneous Exercise On Chapter 10 Solutions 19 Questions
NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra
The chapter Vector Algebra belongs to the unit Vectors and Three – Dimensional Geometry, that adds up to 14 marks of the total marks. There are 4 exercises along with a miscellaneous exercise in this chapter to help students understand the concepts related to Vectors and Vector Algebra clearly. Some of the topics discussed in the tenth Chapter of NCERT Solutions for Class 12 Maths are as follows:
- The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.
- The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as l=(a/r), m=(b/r) n=(c/r)
- The vector sum of the three sides of a triangle taken in order is 0.
- The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
- The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) according as the value of λ is positive (or negative).
The other concepts and topics explained in the chapter can be understood by going through the Chapter 10 of the NCERT textbook for class 12.
Key Features of NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra
Studying the Vector Algebra of Class 12 enables the students to understand the following:
Vectors and scalars, magnitude and direction of a vector.Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.