Class 12 Maths Chapter 10 Vector Algebra – Exercise 10.2 NCERT Solutions

Introduction

Exercise 10.2 focuses on addition, subtraction, and multiplication of vectors by scalars. Students learn how to perform basic vector operations, represent them geometrically, and apply properties such as commutativity and associativity. This exercise builds the foundation for dot product and cross product in later sections.

Key Concepts

1. Vector Addition: If $\vec{a}=(a_1,a_2,a_3)$ and $\vec{b}=(b_1,b_2,b_3)$, then

$$\vec{a}+\vec{b}=(a_1+b_1,a_2+b_2,a_3+b_3)$$

2. Vector Subtraction:

$$\vec{a}-\vec{b}=(a_1-b_1,a_2-b_2,a_3-b_3)$$

3. Scalar Multiplication: If $k$ is scalar,

$$k\vec{a}=(k{a}_1,k{a}_2,k{a}_3)$$

4. Properties:

   • Commutative: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$

   • Associative: $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$

   • Distributive: $k(\vec{a}+\vec{b})=k\vec{a}+k\vec{b}$

Students Frequently Make Mistakes

• Forgetting to add/subtract corresponding components.

• Errors in scalar multiplication.

• Confusing magnitude with components.

• Skipping vector notation in answers.

• Misinterpreting geometric meaning of addition.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find $(2\widehat{i}+3\widehat{j})+(4\widehat{i}-\widehat{j})$.

$$=6\widehat{i}+2\widehat{j}$$

Q2. Find $(3\widehat{i}-2\widehat{j}+4\widehat{k})-(\widehat{i}+5\widehat{j}-2\widehat{k})$.

$$=(3-1)\widehat{i}+(-2-5)\widehat{j}+(4-(-2))\widehat{k}=2\widehat{i}-7\widehat{j}+6\widehat{k}$$

Q3. Find $2(3\widehat{i}+4\widehat{j}-\widehat{k})$.

$$=6\widehat{i}+8\widehat{j}-2\widehat{k}$$

Q4. Find $(-2)(\widehat{i}-3\widehat{j}+2\widehat{k})$.

$$=-2\widehat{i}+6\widehat{j}-4\widehat{k}$$

Q5. Find $(\widehat{i}+2\widehat{j})+(\widehat{j}+3\widehat{k})$.

$$=\widehat{i}+3\widehat{j}+3\widehat{k}$$

Q6. Find $(2\widehat{i}+3\widehat{j}+4\widehat{k})-(\widehat{i}+2\widehat{j}+3\widehat{k})$.

$$=(2-1)\widehat{i}+(3-2)\widehat{j}+(4-3)\widehat{k}=\widehat{i}+\widehat{j}+\widehat{k}$$

Q7. Find $3(\widehat{i}-\widehat{j}+\widehat{k})$.

$$=3\widehat{i}-3\widehat{j}+3\widehat{k}$$

Q8. Find $(4\widehat{i}+5\widehat{j})+(-3\widehat{i}+2\widehat{j})$.

$$=(4-3)\widehat{i}+(5+2)\widehat{j}=\widehat{i}+7\widehat{j}$$

Q9. Find $(2\widehat{i}-\widehat{j}+3\widehat{k})-(\widehat{i}+2\widehat{j}+4\widehat{k})$.

$$=(2-1)\widehat{i}+(-1-2)\widehat{j}+(3-4)\widehat{k}=\widehat{i}-3\widehat{j}-\widehat{k}$$

Q10. Find $5(\widehat{i}+2\widehat{j}+3\widehat{k})$.

$$=5\widehat{i}+10\widehat{j}+15\widehat{k}$$

FAQs (10)

FAQ1. What is vector addition? Adding corresponding components of vectors.

FAQ2. What is vector subtraction? Subtracting corresponding components of vectors.

FAQ3. What is scalar multiplication? Multiplying each component by scalar.

FAQ4. What is commutative property? $\vec{a}+\vec{b}=\vec{b}+\vec{a}$.

FAQ5. What is associative property? $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$.

FAQ6. What is distributive property? $k(\vec{a}+\vec{b})=k\vec{a}+k\vec{b}$.

FAQ7. What is position vector? Vector from origin to point.

FAQ8. What is unit vector? Vector with magnitude 1.

FAQ9. Why use vector operations? To represent physical quantities like force, velocity.

FAQ10. Why is Exercise 10.2 important? It builds foundation for dot and cross product.

Conclusion

Exercise 10.2 has 10 solved questions and 10 FAQs that strengthen your understanding of vector addition, subtraction, and scalar multiplication. This builds the foundation for advanced vector operations in Class 12 Maths.

visit:www.fuzymathacademy.com