Class 12 Maths Chapter 10 Vector Algebra – Exercise 10.4 NCERT Solutions

Introduction

Exercise 10.4 focuses on the vector (cross) product of vectors. Students learn how to calculate cross products, understand their geometric meaning, and apply properties to solve problems involving perpendicularity, area of parallelogram/triangle, and direction of vectors. This exercise is fundamental for applications in physics, mechanics, and geometry.

Formula Used

1. Cross 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

$$\vec{a}\times \vec{b}=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\mid$$

2. Geometric Meaning:

$$\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }=\mathrm{\mid }\vec{a}\mathrm{\mid }\mathrm{\mid }\vec{b}\mathrm{\mid }\sin \theta$$

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

3. Area of Parallelogram:

$$A=\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }$$

4. Area of Triangle:

$$A=\frac{1}{2}\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }$$

Students Frequently Make Mistakes

• Forgetting determinant expansion order.

• Confusing cross product with dot product.

• Errors in sign convention.

• Misinterpreting geometric meaning (direction vs magnitude).

• Skipping unit vector representation.

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

Q1. Find $(\widehat{i}+\widehat{j})\times (\widehat{i}-\widehat{j})$.

$$=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 1& 0\\ 1& -1& 0\end{array}\mid =(0-0)\widehat{i}-(0-0)\widehat{j}+(-1-1)\widehat{k}=-2\widehat{k}$$

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

$$=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 2& 3\\ 2& 1& 1\end{array}\mid =(2-3)\widehat{i}-(1-6)\widehat{j}+(1-4)\widehat{k}=-\widehat{i}+5\widehat{j}-3\widehat{k}$$

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

$$=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 2& -1& 1\\ 1& 3& -2\end{array}\mid =(-2+3)\widehat{i}-(-4-1)\widehat{j}+(6+1)\widehat{k}=\widehat{i}+5\widehat{j}+7\widehat{k}$$

Q4. Find $\widehat{i}\times \widehat{j}$.

$$=\widehat{k}$$

Q5. Find $\widehat{j}\times \widehat{k}$.

$$=\widehat{i}$$

Q6. Find $\widehat{k}\times \widehat{i}$.

$$=\widehat{j}$$

Q7. Find $\widehat{i}\times \widehat{i}$.

$$=0$$

Q8. Find area of parallelogram formed by vectors $\vec{a}=\widehat{i}+2\widehat{j}$, $\vec{b}=2\widehat{i}+3\widehat{j}$.

$$\vec{a}\times \vec{b}=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 2& 0\\ 2& 3& 0\end{array}\mid =(0-0)\widehat{i}-(0-0)\widehat{j}+(3-4)\widehat{k}=-\widehat{k}$$

$$\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }=1\Rightarrow \text{Area}=1$$

Q9. Find area of triangle formed by vectors $\vec{a}=\widehat{i}+2\widehat{j}+3\widehat{k}$, $\vec{b}=2\widehat{i}+3\widehat{j}+4\widehat{k}$.

$$\vec{a}\times \vec{b}=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 2& 3\\ 2& 3& 4\end{array}\mid =(8-9)\widehat{i}-(4-6)\widehat{j}+(3-4)\widehat{k}=-\widehat{i}+2\widehat{j}-\widehat{k}$$

$$\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }=\sqrt{1+4+1}=\sqrt{6}$$

$$\text{Area of triangle}=\frac{1}{2}\sqrt{6}$$

Q10. Find angle between vectors $\vec{a}=\widehat{i}+2\widehat{j}$, $\vec{b}=2\widehat{i}+4\widehat{j}$. Since $\vec{b}=2\vec{a}$, they are parallel.

$$\theta =0^{\circ }$$

FAQs (10)

FAQ1. What is cross product? Vector product of two vectors.

FAQ2. How to calculate cross product? Using determinant expansion.

FAQ3. What is geometric meaning of cross product? Magnitude gives area, direction is perpendicular.

FAQ4. What is $\widehat{i}\times \widehat{j}$? $\widehat{k}$.

FAQ5. What is $\widehat{j}\times \widehat{k}$? $\widehat{i}$.

FAQ6. What is $\widehat{k}\times \widehat{i}$? $\widehat{j}$.

FAQ7. What is $\widehat{i}\times \widehat{i}$?

FAQ8. What is area of parallelogram formed by two vectors? $\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }$.

FAQ9. What is area of triangle formed by two vectors? $\frac{1}{2}\mathrm{\mid }\vec{a}\times \vec{b}\mathrm{\mid }$.

FAQ10. Why is Exercise 10.4 important? It builds mastery of vector cross product and geometric applications.

Conclusion

Exercise 10.4 has 10 solved questions and 10 FAQs that strengthen your understanding of vector cross product. This builds the foundation for advanced vector operations in Class 12 Maths.

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