Class 12 Maths Chapter 10 Vector Algebra – Exercise 10.1 NCERT Solutions

Introduction

Exercise 10.1 introduces the basic concepts of vectors. Students learn about magnitude, direction, representation, and equality of vectors. This exercise builds the foundation for vector operations such as addition, subtraction, and scalar multiplication.

Key Concepts

1. Vector Definition: A quantity with both magnitude and direction.

2. Magnitude of Vector: If $\vec{a}=(x,y,z)$, then

$$\mathrm{\mid }\vec{a}\mathrm{\mid }=\sqrt{x^2+y^2+z^2}$$

3. Equality of Vectors: Two vectors are equal if they have same magnitude and direction.

4. Position Vector: Vector representing position of point $P(x,y,z)$ relative to origin:

$$\overrightarrow{OP}=(x,y,z)$$

Students Frequently Make Mistakes

• Confusing scalar and vector quantities.

• Forgetting to apply square root in magnitude formula.

• Misinterpreting direction ratios.

• Errors in representing position vectors.

• Skipping unit vector normalization.

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

Q1. Find magnitude of vector $\vec{a}=3\widehat{i}+4\widehat{j}$.

$$\mathrm{\mid }\vec{a}\mathrm{\mid }=\sqrt{3^2+4^2}=5$$

Q2. Find magnitude of vector $\vec{b}=2\widehat{i}-\widehat{j}+2\widehat{k}$.

$$\mathrm{\mid }\vec{b}\mathrm{\mid }=\sqrt{2^2+(-1{)}^2+2^2}=\sqrt{9}=3$$

Q3. Find unit vector in direction of $\vec{a}=6\widehat{i}-8\widehat{j}$.

$$\mathrm{\mid }\vec{a}\mathrm{\mid }=\sqrt{36+64}=10$$

$$\widehat{a}=\frac{\vec{a}}{\mathrm{\mid }\vec{a}\mathrm{\mid }}=\frac{6}{10}\widehat{i}-\frac{8}{10}\widehat{j}=\frac{3}{5}\widehat{i}-\frac{4}{5}\widehat{j}$$

Q4. Find position vector of point $P(2,3,6)$.

$$\overrightarrow{OP}=2\widehat{i}+3\widehat{j}+6\widehat{k}$$

Q5. Find magnitude of vector $\vec{c}=-\widehat{i}+2\widehat{j}-2\widehat{k}$.

$$\mathrm{\mid }\vec{c}\mathrm{\mid }=\sqrt{(-1{)}^2+2^2+(-2{)}^2}=\sqrt{9}=3$$

Q6. Find unit vector in direction of $\vec{d}=4\widehat{i}+3\widehat{j}$.

$$\mathrm{\mid }\vec{d}\mathrm{\mid }=\sqrt{16+9}=5$$

$$\widehat{d}=\frac{4}{5}\widehat{i}+\frac{3}{5}\widehat{j}$$

Q7. Find magnitude of vector $\vec{e}=7\widehat{i}-24\widehat{j}$.

$$\mathrm{\mid }\vec{e}\mathrm{\mid }=\sqrt{49+576}=\sqrt{625}=25$$

Q8. Find unit vector in direction of $\vec{f}=2\widehat{i}+2\widehat{j}+1\widehat{k}$.

$$\mathrm{\mid }\vec{f}\mathrm{\mid }=\sqrt{4+4+1}=\sqrt{9}=3$$

$$\widehat{f}=\frac{2}{3}\widehat{i}+\frac{2}{3}\widehat{j}+\frac{1}{3}\widehat{k}$$

Q9. Find position vector of point $Q(-3,4,-5)$.

$$\overrightarrow{OQ}=-3\widehat{i}+4\widehat{j}-5\widehat{k}$$

Q10. Find magnitude of vector $\vec{g}=5\widehat{i}+12\widehat{j}$.

$$\mathrm{\mid }\vec{g}\mathrm{\mid }=\sqrt{25+144}=\sqrt{169}=13$$

FAQs (10)

FAQ1. What is vector? Quantity with magnitude and direction.

FAQ2. What is scalar? Quantity with only magnitude.

FAQ3. How to find magnitude of vector? $\sqrt{x^2+y^2+z^2}$.

FAQ4. What is unit vector? Vector with magnitude 1.

FAQ5. How to find unit vector? Divide vector by its magnitude.

FAQ6. What is position vector? Vector from origin to point.

FAQ7. What is equality of vectors? Same magnitude and direction.

FAQ8. What is difference between scalar and vector? Scalar has only magnitude, vector has both.

FAQ9. What is magnitude of $3\widehat{i}+4\widehat{j}$?

FAQ10. Why is Exercise 10.1 important? It builds foundation for vector operations in algebra and geometry.

Conclusion

Exercise 10.1 has 10 solved questions and 10 FAQs that strengthen your understanding of basic vector concepts. This begins the Vector Algebra chapter in Class 12 Maths.

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