NCERT Solutions for Class 12 Maths Chapter 10 – Free PDF Download
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 students score high marks not only in board exams but also in competitive exams. The NCERT Solutions for Class 12 Maths are important to prepare for the various problems asked during the Class 12 Maths board examination.
These Class 12 Maths NCERT Solutions for Vector Algebra are very easytounderstand and are in accordance with the latest CBSE Syllabus 202324. Solving through different exercises and problem sets gives students the confidence to perform better in the CBSE exams. Students can also avail of NCERT Solutions for Class 12 Maths Chapter 10 in a downloadable PDF for free to practise them offline as well.
NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra
NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra
Exercise 10.1 Page No: 428
1. Represent graphically a displacement of 40 km, 30Â° east of north.
Solution:
The vector
represents the displacement of 40 km, 30^{o} east of north.
2. Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres northwest (iii) 40Â°
(iv) 40 watt (v) 10^{â€“19}Â coulomb (vi) 20 m/s^{2}
Solution:
(i) 10 kg is a scalar quantity because it has only magnitude.
(ii) 2 meters northwest is a vector quantity as it has both magnitude and direction.
(iii) 40Â° is a scalar quantity as it has only magnitude.
(iv) 40 watt is a scalar quantity as it has only magnitude.
(v) 10^{â€“19}Â coulomb is a scalar quantity as it has only magnitude.
(vi) 20 m/s^{2}Â 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 Â andare collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having the same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Solution:
(i) True.
Vectors
Â 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 vectors 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 the same magnitude.
Solution:
3. Write two different vectors having the same direction.
Solution:
Two different vectors having the same directions are
Let us
4. Find the values ofÂ xÂ andÂ yÂ so that the vectorsÂ are equal
Solution:
Given vectors
will be equal only if their corresponding components are equal.
Thus, the required values ofÂ xÂ andÂ y are 2 and 3, respectively.
5. Find the scalar and vector components of the vector with the initial point (2, 1) and terminal point (â€“5, 7).
Solution:
The scalar and vector components are
The vector with initial point P (2, 1) and terminal point Q (â€“5, 7) can be shown as
Thus, the required scalar components are â€“7 and 6, while the vector components are
6. Find the sum of the vectors.
Solution:
Let us find the sum of the vectors.
7. Find the unit vector in the direction of the vector.
Solution:
We know that
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:
We know that,
9. For given vectors andÂ ,Â find the unit vector in the direction of the vectorÂ
Solution:
We know that,
10. Find a vector in the direction of the vector which has magnitude of 8 units.
Solution:
Firstly,
11. Show that the vectorsare collinear.
Solution:
First,
Therefore, we can say that the given vectors are collinear.
12. Find the direction cosines of the vectorÂ
Solution:
First,
13. Find the direction cosines of the vector joining the points A (1, 2, â€“3) and
B (â€“1, â€“2, 1) directed from A to B.
Solution:
We know that the given points are A (1, 2, â€“3) and B (â€“1, â€“2, 1).
Now,
14. Show that the vectorÂ is equally inclined to the axes OX, OY, and OZ.
Solution:
Firstly,
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 ratio 2:1
(i) internally
(ii) externally
Solution:
We know that
The position vector of point R dividing the line segment joining two points.
P and Q in the ratioÂ m:Â nÂ is given by
16. Find the position vector of the midpoint of the vector joining the points P (2, 3, 4) and Q (4, 1, â€“ 2).
Solution:
The position vector of midpoint R of the vector joining points P (2, 3, 4) and Q (4, 1, â€“ 2) is given by
17. Show that the points A, B and C with position vectors,,Â respectively, form the vertices of a rightangled triangle.
Solution:
We know
Given position vectors of points A, B, and C are
Hence, proved that the given points form the vertices of a rightangled triangle.
18. In triangle ABC (Fig 10.18), which of the following is not true.
Solution:
First, let us consider,
19. IfÂ are two collinear vectors, then which of the following is incorrect?
A.Â , for some scalar Î»
B.Â
C. The respective components of are proportional
D. Both the vectors have the same direction, but different magnitudes
Solution:
We know,
Exercise 10.3 Page No: 447
1. Find the angle between two vectorsandwith magnitudes âˆš3 and 2, respectively having.
Solution:
First, let us consider,
2. Find the angle between the vectors
Solution:
Let us consider the
Hence, the angle between the vectors is cos^{1} (5/7).
3. Find the projection of the vectoron the vector.
Solution:
First,
4. Find the projection of the vectoron the vector.
Solution:
First,
Hence, the projection is 60/âˆš114.
5. Show that each of the given three vectors is a unit vector.
Also, show that they are mutually perpendicular to each other.
Solution:
It is given that
6. Find
Solution:
Let us consider,
7. Evaluate the product
Solution:
Let us consider the given expression
8. Find the magnitude of two vectors, having the same magnitude and such that the angle between them is 60Â° and their scalar product is Â½.
Solution:
First,
Hence, the magnitude of the two vectors is 1.
Solution:
Let us consider,
Hence, the value is âˆš13.
10. Ifare such thatis perpendicular to, then find the value ofÂ Î».
Solution:
We know that the
11. Show thatÂ is perpendicular to, for any two nonzero vectors.
Solution:
Let us consider,
12. If, then what can be concluded about the vector?
Solution:
We know,
13. IfÂ are unit vectors such thatÂ , find the value ofÂ .
Solution:
Consider the given vectors,
Hence, the value is 3/2.
14. If either vector, then. But the converse need not be true. Justify your answer with an example.
Solution:
First,
15. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (â€“1, 0, 0), (0, 1, 2), respectively, then find âˆ ABC. [âˆ ABC is the angle between the vectorsand]
Solution:
We know,
Hence, the angle is cos^{1} (10/ âˆš102).
16. Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, â€“1) are collinear.
Solution:
Let us consider,
Given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, â€“1).
Now,
Therefore, the given points A, B, and C are collinear.
17. Show that the vectorsform the vertices of a rightangled triangle.
Solution:
First, consider
Solution:
Explanation:
Exercise 10.4 Page No: 454
1. Find, ifÂ and
Solution:
It is given that,
2. Find a unit vector perpendicular to each of the vectorÂ and, whereÂ and.
Solution:
It is given that,
Solution:
First,
4. Show that
Solution:
First, consider the LHS,
We have
5. FindÂ Î»Â andÂ Î¼Â ifÂ .
Solution:
It is given that
Solution:
It is given that
Solution:
First, let us consider,
9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Solution:
We know
10. Find the area of the parallelogram whose adjacent sides are determined by the vectorÂ .
Solution:
Let us consider,
Solution:
Explanation:
12. Area of a rectangle having vertices A, B, C, and D with position vectorsÂ andÂ , respectively is
Solution:
Explanation:
Miscellaneous Exercise Page No: 458
1. Write down a unit vector in XYplane, making an angle of 30Â° with the positive direction of the xaxis.
Solution:
Let us consider,
2. Find the scalar components and magnitude of the vector joining the points P (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}).
Solution:
First, let us consider,
3. A girl walks 4 km towards west, then she walks 3 km in a direction 30Â° east of north and stops. Determine the girlâ€™s displacement from her initial point of departure.
Solution:
It is given that
Let O and B be the initial and final positions of the girl, respectively.
Then, the girlâ€™s position can be shown as
4. If, then, is it true that? Justify your answer.
Solution:
It is given that,
5. Find the value ofÂ xÂ for whichis a unit vector.
Solution:
We know,
6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
.
Solution:
Let us consider that the
7. If, find a unit vector parallel to the vector.
Solution:
Let us consider the given vectors,
8. Show that the points A (1, â€“2, â€“8), B (5, 0, â€“2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Solution:
First, let us consider,
9. Find the position vector of a point R, which divides the line joining two points P and Q, whose position vectors areexternally in the ratio 1: 2. Also, show that P is the midpoint of the line segment RQ.
Solution:
We know,
10. The two adjacent sides of a parallelogram areandÂ .
Find the unit vector parallel to its diagonal. Also, find its area.
Solution:
First, let us consider,
11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are.
Solution:
First,
Letâ€™s assume a vector to be equally inclined to axes OX, OY, and OZ at angleÂ Î±.
Then, the direction cosines of the vector are cosÂ Î±, cosÂ Î±, and cosÂ Î±.
Now, we know that
Therefore, the direction cosines of the vector, which are equally inclined to the axes, are
Hence, proved.
Solution:
Assume,
13. The scalar product of the vectorwith a unit vector along the sum of vectorsÂ andÂ is equal to one. Find the value of.
Solution:
Letâ€™s consider the
14. IfÂ are mutually perpendicular vectors of equal magnitudes, show that the vectorÂ is equally inclined toÂ and.
Solution:
Let’s assume,
Hence proved.
15. Prove that, if and only ifÂ are perpendicular, given.
Solution:
It is given that
Hence, proved.
Solution:
Explanation:
Solution:
Explanation:
Hence the correct answer is D.
18. The value ofÂ is
(A) 0 (B) â€“1 (C) 1 (D) 3
Solution:
Explanation:
It is given that,
Hence, the correct answer is C.
Solution:
Explanation:
Also AccessÂ 
NCERT Exemplar for Class 12 Maths Chapter 10 
CBSE Notes for Class 12 Maths Chapter 10 
The major concepts of Maths covered in Chapter 10 – Vector Algebra of NCERT Solutions for Class 12 include
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
Access NCERT Solutions for Class 12 Maths Chapter 10 exercisewise from the below links
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 ThreeDimensional Geometry, which 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 Î» changes the magnitude of the vector by the multiple Î» and keeps the direction the same (or makes it opposite) accordingly, as the value of Î» is positive (or negative).
The other concepts and topics explained in the NCERT Solutions of this chapter can be understood by going through 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 using the NCERT Solutions 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, the 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, the scalar triple product of vectors. Students can utilise the NCERT Solutions for Class 12 Maths Chapter 10 for any quick reference to comprehend complex topics.
Disclaimer –Â
Dropped Topics –Â 10.7 Scalar Triple Product, 10.7.1 Coplanarity of Three Vectors
Frequently Asked Questions on NCERT Solutions for Class 12 Maths Chapter 10
How many questions are present in each exercise of NCERT Solutions for Class 12 Maths Chapter 10?
Exercise 10.1 â€“ 5 Questions
Exercise 10.2 â€“ 19 Questions
Exercise 10.3 â€“ 18 Questions
Exercise 10.4 â€“ 12 Questions
Miscellaneous Exercise â€“ 19 Questions
What are the major concepts discussed in NCERT Solutions for Class 12 Maths Chapter 10?
10.1 â€“ Introduction
10.2 â€“ Basic Concepts
10.3 â€“ Types of Vectors
10.4 â€“ Addition of Vectors
10.5 â€“ Multiplication of a vector by a scalar
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