NCERT Solutions for Class 12 Mathematics Chapter Vector Algebra

This worksheet is designed to help Class 12 students understand the important concepts of Mathematics Chapter Vector Algebra in a clear and structured way. This chapter focuses on vectors, their types, operations, and applications such as dot product, cross product, and geometric interpretations. It is an essential topic in higher mathematics and helps build a strong foundation for physics and engineering concepts. This worksheet provides complete and accurate NCERT Solutions, strictly aligned with the chapter content, helping students practice and revise effectively.

Chapter summary: stories, poems & themes

This chapter is concept-based and does not include stories or poems. It focuses on mathematical understanding of vectors and their applications. Students learn about scalar and vector quantities, direction, magnitude, and operations such as addition, subtraction, dot product, and cross product. The chapter also includes problem-solving related to geometry, direction cosines, projections, and applications like work done and area calculations.

What this NCERT chapter covers?

• Understanding scalar and vector quantities  
• Representation of vectors and direction  
• Operations on vectors such as addition and subtraction  
• Magnitude and unit vectors  
• Dot product and its applications  
• Cross product and its geometric meaning  
• Direction cosines and position vectors  
• Application-based problem solving  

How to use these NCERT solutions?

Students should first attempt all questions from the worksheet on their own to build problem-solving skills. After attempting, they can refer to these NCERT Solutions to verify answers and understand the correct steps. Parents and teachers can use this worksheet to guide students and track their progress. The solutions follow the exact NCERT order and structure, making revision simple and effective.

Student tips & learning tricks

• Always check whether a quantity is scalar or vector before solving  
• Practice writing vectors in component form clearly  
• Be careful while calculating magnitudes and square roots  
• Learn formulas for dot product and cross product properly  
• Avoid sign errors in vector operations  
• Practice diagrams for better understanding of direction  

Why NCERT solutions are important?

NCERT Solutions are important because they follow the exact syllabus and exam pattern. They help students build strong conceptual clarity and improve accuracy in solving problems. These solutions boost confidence, support revision, and prepare students for board exams and competitive exams effectively.

Complete answer key – NCERT solutions

Exercise No. 10.1

1 A vector of magnitude 40 km making an angle of 30° towards East from North.

2  
(i) 10 kg → Scalar  
Reason: Only magnitude, no direction  
(ii) 2 meters north-west → Vector  
Reason: Has magnitude and direction  
(iii) 40° → Scalar  
Reason: Only magnitude  
(iv) 40 watt → Scalar  
Reason: Only magnitude  
(v) 10⁻¹⁹ coulomb → Scalar  
Reason: Only magnitude  
(vi) 20 m/s² → Vector  
Reason: Acceleration has direction  

3  
(i) Time period → Scalar  
Reason: Only magnitude  
(ii) Distance → Scalar  
Reason: No direction  
(iii) Force → Vector  
Reason: Has magnitude and direction  
(iv) Velocity → Vector  
Reason: Has magnitude and direction  

Exercise No. 10.2

1  
(v) Work done → Scalar  
Reason: No direction  

2  
(i) Coinitial vectors: Vectors starting from same point Example: AB, AD  
(ii) Equal vectors: Vectors having same magnitude and direction Example: AB = DC  
(iii) Collinear but not equal: Vectors parallel but different magnitude/direction Example: AB and DC  

3  
(i) a and –a are collinear → TRUE  
(ii) Two collinear vectors are always equal in magnitude → FALSE  
(iii) Two vectors having same magnitude are collinear → FALSE  
(iv) Two collinear vectors having same magnitude are equal → FALSE  

4  
Answer: (2/√13)i + (3/√13)j  

5  
Answer: (–3/7)i + (6/7)j + (2/7)k  

6  
Answer: 5  

7  
Answer: (7/3)i + (14/3)j + (14/3)k  

8  
Answer: (3/√50)i + (4/√50)j + (5/√50)k  

9  
Answer: (10/3)i – (5/3)j + (10/3)k  

10  
Answer: 2b – a  

11  
Answer: (2b + a)/3  

12  
Answer: (1/√2)i – (1/√2)j  

13  
Answer: (40/√30)i – (16/√30)j + (8/√30)k  

14  
Answer: (2/√14, 3/√14, 1/√14)  

15  
Answer: Vectors are collinear  

16  
Answer: –2i –4j + 4k  

17  
Answer: Vector is equally inclined to axes  

18  
Answer: Internal: j + k External: –4i + j + k  

19  
Answer: 3i + 2j + k  

Exercise No. 10.3

1 Answer: 8  

2 Answer: 9  

3 Answer: 13  

4 Answer: 90°  

5 Answer: 90°  

6 Answer: 5/√2  

7 Answer: (5/2)i + (5/2)j  

8 Answer: 4 units  

9 Answer: Not perpendicular  

10 Answer: θ = cos⁻¹(–1/√50)  

11 Answer: 24/5  

12 Answer: (96/25)i + (72/25)j  

13 Answer: 60°  

14 Answer: θ = cos⁻¹(1/3)  

15 Answer: a·b = 0  

16 Answer: θ = 0°  

17 Answer: θ = 180°  

18 Answer: λ = –1/2  

Exercise No. 10.4

1 Answer: –i + 5j – 3k  

2 Answer: –2i – j + 3k  

3 Answer: –2k  

4 Answer: √6  

5 Answer: √6 / 2  

6 Answer: (1/√6)i – (2/√6)j + (1/√6)k  

7 Answer: Vectors are parallel  

8 Answer: 5  

9 Answer: 5/2  

10 Answer: i + 7j – 5k  

11 Answer: √6  

12 Answer: (1/√6)i – (2/√6)j + (1/√6)k  

Miscellaneous Exercise on Chapter 10

1 Answer: 90°  

2 Answer: (96/25)i + (72/25)j  

3 Answer: (2/3, –2/3, 1/3)  

4 Answer: Vectors are collinear  

5 Answer: √6 / 2  

6 Answer: λ = –1/2  

7 Answer: (10/3)i + (5/3)j + (10/3)k  

8 Answer: θ = cos⁻¹(1/3)  

9 Answer: 5/√2  

10 Answer: i + 7j – 5k  

11 Answer: 5√3  

12 Answer: Vectors are parallel  

13 Answer: 5√3  

14 Answer: (5√3)/2  

15 Answer: θ = cos⁻¹[38 / √1450]  

16 Answer: (33/14)i + (11/14)j + (22/14)k  

17 Answer: 11/√14  

18 Answer: (1/5√3)i + (7/5√3)j – (5/5√3)k  

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