Class 12 Maths Chapter 10 Vector Algebra – Exercise 10.3 NCERT Solutions

Introduction

Exercise 10.3 focuses on the scalar (dot) product of vectors. Students learn how to calculate dot products, apply geometric interpretation, and use properties to solve problems involving angles between vectors and orthogonality. This exercise is fundamental for applications in physics, engineering, and geometry.

Formula Used

1. Dot Product Definition: If $\vec{a}=a_1\widehat{i}+a_2\widehat{j}+a_3\widehat{k}$ and $\vec{b}=b_1\widehat{i}+b_2\widehat{j}+b_3\widehat{k}$, then

$$\vec{a}\cdot \vec{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$$

2. Geometric Meaning:

$$\vec{a}\cdot \vec{b}=\mathrm{\mid }\vec{a}\mathrm{\mid }\mathrm{\mid }\vec{b}\mathrm{\mid }\cos \theta$$

where $\theta$ is angle between $\vec{a}$ and $\vec{b}$.

3. Orthogonality: If $\vec{a}\cdot \vec{b}=0$, then vectors are perpendicular.

Students Frequently Make Mistakes

• Forgetting to multiply corresponding components correctly.

• Confusing dot product with cross product.

• Errors in calculating magnitude before applying formula.

• Misinterpreting orthogonality condition.

• Skipping angle interpretation.

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

Q1. Find $(\widehat{i}+2\widehat{j}+3\widehat{k})\cdot (2\widehat{i}+\widehat{j}+\widehat{k})$.

$$=1\cdot 2+2\cdot 1+3\cdot 1=2+2+3=7$$

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

$$=2\cdot 1+(-1)\cdot 3+1\cdot (-2)=2-3-2=-3$$

Q3. Find angle between $\vec{a}=\widehat{i}$ and $\vec{b}=\widehat{j}$.

$$\vec{a}\cdot \vec{b}=0\Rightarrow \theta ={90}^{\circ }$$

Q4. Find angle between $\vec{a}=\widehat{i}+\widehat{j}$ and $\vec{b}=\widehat{i}-\widehat{j}$.

$$\vec{a}\cdot \vec{b}=1\cdot 1+1\cdot (-1)=0\Rightarrow \theta ={90}^{\circ }$$

Q5. Find $(\widehat{i}+2\widehat{j})\cdot (3\widehat{i}+4\widehat{j})$.

$$=1\cdot 3+2\cdot 4=3+8=11$$

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

$$=2\cdot 1+3\cdot (-1)+4\cdot 1=2-3+4=3$$

Q7. Find angle between $\vec{a}=\widehat{i}+2\widehat{j}$ and $\vec{b}=2\widehat{i}+4\widehat{j}$.

$$\vec{a}\cdot \vec{b}=1\cdot 2+2\cdot 4=2+8=10$$

$$\mathrm{\mid }\vec{a}\mathrm{\mid }=\sqrt{1^2+2^2}=\sqrt{5},\mathrm{\mid }\vec{b}\mathrm{\mid }=\sqrt{2^2+4^2}=\sqrt{20}=2\sqrt{5}$$

$$\cos \theta =\frac{10}{\sqrt{5}\cdot 2\sqrt{5}}=\frac{10}{10}=1\Rightarrow \theta =0^{\circ }$$

Q8. Find angle between $\vec{a}=\widehat{i}+2\widehat{j}+2\widehat{k}$ and $\vec{b}=2\widehat{i}+4\widehat{j}+4\widehat{k}$. Clearly, $\vec{b}=2\vec{a}$. $\theta =0^{\circ }$.

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

$$=1\cdot 1+2\cdot (-2)+3\cdot 3=1-4+9=6$$

Q10. Find angle between $\vec{a}=\widehat{i}+2\widehat{j}+3\widehat{k}$ and $\vec{b}=2\widehat{i}+4\widehat{j}+6\widehat{k}$. $\vec{b}=2\vec{a}$. $\theta =0^{\circ }$.

FAQs (10)

FAQ1. What is dot product? Scalar product of two vectors.

FAQ2. How to calculate dot product? Multiply corresponding components and add.

FAQ3. What is geometric meaning of dot product? $\mathrm{\mid }\vec{a}\mathrm{\mid }\mathrm{\mid }\vec{b}\mathrm{\mid }\cos \theta$.

FAQ4. When are vectors perpendicular? If dot product = 0.

FAQ5. What is dot product of $\widehat{i}$ and $\widehat{j}$?

FAQ6. What is dot product of $\widehat{i}$ and $\widehat{i}$?

FAQ7. What is angle between $\widehat{i}$ and $\widehat{j}$? 90°.

FAQ8. What is angle between parallel vectors? 0°.

FAQ9. Why is dot product scalar? Because result is a number, not a vector.

FAQ10. Why is Exercise 10.3 important? It builds foundation for angles and orthogonality in vector algebra.

Conclusion

Exercise 10.3 has 10 solved questions and 10 FAQs that strengthen your understanding of scalar (dot) product of vectors. This builds the foundation for advanced vector operations in Class 12 Maths.

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