NCERT Solutions for Class 12 Mathematics Vector Algebra Chapter 10

NCERT solutions for Class 12 Mathematics Chapter Vector Algebra – complete answers & explanations

Vector Algebra is an important chapter in Class 12 Mathematics that introduces students to the concept of vectors, their properties, and operations in three-dimensional space. This chapter helps students understand quantities that have both magnitude and direction, which are widely used in physics, engineering, and real-life applications. From basic definitions to advanced concepts like scalar product and vector product, the chapter builds a strong mathematical foundation. These NCERT solutions provide clear and reliable answers that help students understand the correct approach to solving problems. Students can strengthen their preparation by practicing regularly. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance.

What this NCERT chapter covers?

1. Understanding the concept of vectors and how they differ from scalar quantities.  
2. Learning about position vectors and how to represent points in three-dimensional space.  
3. Studying different types of vectors such as zero vector, unit vector, and collinear vectors.  
4. Exploring operations on vectors like addition, subtraction, and scalar multiplication.  
5. Understanding direction cosines and direction ratios of vectors.  
6. Learning how to find the vector joining two points and its magnitude.  
7. Applying section formula to divide a line segment internally and externally.  
8. Understanding scalar (dot) product and its properties.  
9. Learning about vector (cross) product and its geometric interpretation.  
10. Solving problems related to angles between vectors and projections.  
11. Applying vector concepts to geometry problems like triangles and parallelograms.  
12. Developing problem-solving skills useful for board exams and competitive exams.

How to use these NCERT solutions?

1. Start by attempting each question from the worksheet on your own before checking the answers.  
2. Carefully compare your answers with the provided solutions to identify mistakes.  
3. Focus on understanding the method used rather than memorizing answers.  
4. Practice similar questions to strengthen your concepts.  
5. Use these solutions as a reference while revising before exams.  
6. Parents and teachers can guide students by discussing difficult questions.  
7. Follow the sequence of questions exactly as given to maintain proper flow.  
8. Revise important formulas and concepts alongside solving questions.  

Important tips & tricks for students

1. Always write vectors clearly using proper notation.  
2. Pay attention to direction and magnitude while solving problems.  
3. Practice diagrams wherever required to improve understanding.  
4. Avoid calculation mistakes in dot and cross products.  
5. Revise formulas like projection, angle between vectors, and magnitude regularly.  
6. Read questions carefully before solving to avoid misinterpretation.  
7. Practice different types of problems for better confidence.  
8. Attempt all questions to ensure complete preparation.  

NCERT solutions – complete answer key

Exercise 10.1

1. Answers may vary.

2.  
(i) Scalar  
(ii) Vector  
(iii) Scalar  
(iv) Scalar  
(v) Scalar  
(vi) Vector  

3.  
(i) Scalar  
(ii) Scalar  
(iii) Vector  
(iv) Vector  
(v) Scalar  

4. Answers may vary.

5.  
(i) True  
(ii) False  
(iii) False  
(iv) True  

Exercise 10.3

1. Find the angle between two vectors with magnitudes 3 and 2, respectively having .

2. Find the angle between the vectors.

3. Find the projection of the vector ˆ ˆi j− on the vector ˆ ˆi j+.

4. Find the projection of the vector ˆˆ ˆ3 7i j k+ + on the vector ˆˆ ˆ7 8i j k− +.

5. Show that each of the given three vectors is a unit vector:
(2 3 6 ), (3 6 2 ), (6 2 3 )  
Also, show that they are mutually perpendicular to each other.

6. Find , if .

7. Evaluate the product .

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 1/2.

9. Find , if for a unit vector , .

10. If are such that is perpendicular to , then find the value of λ.

11. Show that is perpendicular to , for any two nonzero vectors.

12. If , then what can be concluded about the vector ?

13. If are unit vectors such that , find the value of.

14. If either vector . But the converse need not be true. Justify your answer with an example.

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.

16. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

17. Show that the vectors form the vertices of a right angled triangle.

18. If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λ is unit vector if  
(A) λ = 1  
(B) λ = –1  
(C) a = |λ|  
(D) a = 1/|λ|  

Exercise 10.4

1. Find .

2. Find a unit vector perpendicular to each of the vector.

3. If a unit vector makes angles with , find θ and hence the components.

4. Show that.

5. Find λ and µ if.

6. Given that and . What can you conclude about the vectors?

7. Let the vectors be given as. Then show that.

8. If either then. Is the converse true? Justify your answer with an example.

9. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

10. Find the area of the parallelogram whose adjacent sides are determined by the vectors.

11. Let the vectors be such that, then is a unit vector, if the angle between is  
(A) π/6  
(B) π/4  
(C) π/3  
(D) π/2  

12. Area of a rectangle having vertices with position vectors is  
(A) 1/2  
(B) 1  
(C) 2  
(D) 4  

Miscellaneous Exercise on Chapter 10

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

2. Find the scalar components and magnitude of the vector joining the points.

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.

4. If , then is it true that ? Justify your answer.

5. Find the value of x for which (x i + j + k) is a unit vector.

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors.

7. If , find a unit vector parallel to the vector.

8. Show that the points A, B and C are collinear, and find the ratio.

9. Find the position vector of a point R which divides the line joining two points externally in the ratio 1:2.

10. Find the unit vector parallel to its diagonal. Also, find its area.

11. Show that the direction cosines of a vector equally inclined to the axes are ±(1/√3, 1/√3, 1/√3).

12. Let. Find a vector which is perpendicular to both.

13. The scalar product is equal to one. Find the value of λ.

14. If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined.

15. Prove that if and only if are perpendicular.

16. If θ is the angle between two vectors, then  
(A) 0 < θ < π/2  
(B) 0 ≤ θ ≤ π/2  
(C) 0 < θ < π  
(D) 0 ≤ θ ≤ π  

17. Let be two unit vectors and θ is the angle between them. Then is a unit vector if  
(A) π/4  
(B) π/3  
(C) π/2  
(D) 2π/3  

18. The value is  
(A) 0  
(B) –1  
(C) 1  
(D) 3  

19. If θ is the angle between any two vectors, then when θ is equal to  
(A) 0  
(B) π/4  
(C) π/2  
(D) π  

Why NCERT solutions help students?

NCERT solutions play a key role in helping students prepare effectively for exams. They provide clarity in concepts, improve problem-solving skills, and help students understand the correct answering pattern. With regular practice, students gain confidence and perform better in exams.

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