Vector Algebra: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. $|\hat{i}+2\hat{j}-2\hat{k}|$.
2. Unit vector along $3\hat{i}-4\hat{k}$.
3. $\vec{a}\cdot \vec{b}$ for $\hat{i}+2\hat{j}+3\hat{k}$, $2\hat{i}-\hat{j}+\hat{k}$.
4. $\lambda$ if $2\hat{i}+\lambda \hat{j}+\hat{k}\perp \hat{i}-2\hat{j}+3\hat{k}$.
5. Projection of $2\hat{i}+3\hat{j}+2\hat{k}$ on $\hat{i}+2\hat{j}+\hat{k}$.
6. Angle between $\hat{i}$ and $\hat{i}+\hat{j}$.
7. $(\hat{i}+\hat{j})\times (\hat{i}-\hat{j})$.
8. Area of parallelogram on $\hat{i}$ and $\hat{j}+\hat{k}$.
9. Direction cosines of $2\hat{i}-\hat{j}+2\hat{k}$.
10. $p$ if $(2\hat{i}+6\hat{j}+14\hat{k})\times (\hat{i}-p\hat{j}+7\hat{k})=\vec{0}$.

D. Short answer I (2 marks each)

1. Find a unit vector perpendicular to $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$.
2. Find the angle between $\hat{i}+\hat{j}-\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$.
3. $|\vec{a}|=3$, $|\vec{b}|=4$, $\vec{a}\cdot \vec{b}=6$: find $|\vec{a}\times \vec{b}|$.
4. $|\vec{a}|=2$, $|\vec{b}|=3$, $\vec{a}\cdot \vec{b}=4$: find $|\vec{a}-\vec{b}|$.
5. Show $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}-3\hat{j}-5\hat{k}$ are perpendicular.
6. Position vector of the point dividing A(1, 2, 3), B(4, –1, 0) internally in 1 : 2.
7. Area of triangle A(1, 0, 0), B(0, 1, 0), C(0, 0, 1).
8. $\vec{a}$ is a unit vector and $(\vec{x}-\vec{a})\cdot (\vec{x}+\vec{a})=8$: find $|\vec{x}|$.

Rough work

E. Short answer II (3 marks each)

1. $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, $|\vec{a}|=3$, $|\vec{b}|=5$, $|\vec{c}|=7$: find the angle between $\vec{a}$ and $\vec{b}$.
2. Find a vector of magnitude 6 perpendicular to both $2\hat{i}-\hat{j}+2\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$.
3. Find the area of the parallelogram whose diagonals are $3\hat{i}+\hat{j}-2\hat{k}$ and $\hat{i}-3\hat{j}+4\hat{k}$.
4. If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, show that $\vec{a}\perp \vec{b}$.
5. Show A(1, 2, 3), B(2, 3, 1), C(3, 1, 2) form an equilateral triangle and find its area.
6. Find the unit vector along the sum of $2\hat{i}+2\hat{j}-5\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$.

F. Long answer (5 marks each)

1. $\vec{a}=4\hat{i}+5\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-4\hat{j}+5\hat{k}$, $\vec{c}=3\hat{i}+\hat{j}-\hat{k}$. Find $\vec{d}$ perpendicular to both $\vec{a}$ and $\vec{b}$ with $\vec{d}\cdot \vec{c}=21$.
2. Using vectors, prove that the angle in a semicircle is a right angle.

G. HOTS (higher-order thinking)

1. Why is $\vec{a}\cdot \vec{b}$ a scalar but $\vec{a}\times \vec{b}$ a vector?
2. When is $|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|$ for nonzero vectors?
3. Why is $\hat{i}\times \hat{j}=\hat{k}$ but $\hat{j}\times \hat{i}=-\hat{k}$?
4. If $\vec{a}\cdot \vec{b}=\vec{a}\cdot \vec{c}$ with $\vec{a}\ne \vec{0}$, must $\vec{b}=\vec{c}$?

20 MCQs

1. $|3\hat{i}-4\hat{j}|=$
   (a) 7 (b) 1 (c) 5 (d) 25
2. $\hat{i}\cdot \hat{i}+\hat{j}\cdot \hat{j}+\hat{k}\cdot \hat{k}=$
   (a) 3 (b) 0 (c) 1 (d) –3
3. $\hat{i}\times \hat{i}=$
   (a) 1 (b) $\hat{k}$ (c) $\hat{i}$ (d) $\vec{0}$
4. $\hat{k}\times \hat{j}=$
   (a) $\hat{i}$ (b) $-\hat{i}$ (c) $\vec{0}$ (d) $\hat{j}$
5. $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|$ when $\theta =$
   (a) $\frac{\pi }{2}$ (b) 0 (c) $\pi$ (d) $\frac{\pi }{4}$
6. Projection of $\hat{i}+\hat{j}$ on $\hat{i}$
   (a) 0 (b) 2 (c) $\sqrt{2}$ (d) 1
7. Direction cosines of $\hat{i}+\hat{j}+\hat{k}$
   (a) $\frac{1}{\sqrt{3}}$ each (b) 1, 1, 1 (c) $\frac{1}{3}$ each (d) 3, 3, 3
8. Unit vector along $4\hat{i}-3\hat{k}$
   (a) $4\hat{i}-3\hat{k}$ (b) $\frac{1}{7}(4\hat{i}-3\hat{k})$ (c) $\frac{1}{5}(4\hat{i}-3\hat{k})$ (d) $\frac{1}{25}(4\hat{i}-3\hat{k})$
9. $\lambda$ for $2\hat{i}-\hat{j}+\lambda \hat{k}\perp \hat{i}+\hat{j}+\hat{k}$
   (a) 1 (b) 2 (c) 0 (d) –1
10. Area of parallelogram on $\hat{i}$ and $\hat{j}$
    (a) 0 (b) 1 (c) 2 (d) $\frac{1}{2}$
11. $|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|$ when $\theta =$
    (a) 0 (b) $\pi$ (c) $\frac{\pi }{2}$ (d) $\frac{\pi }{4}$
12. $|\vec{a}|=2$, $|\vec{b}|=3$, $\theta =60^{\circ }$: $\vec{a}\cdot \vec{b}=$
    (a) 3 (b) 6 (c) $3\sqrt{3}$ (d) 1
13. $\hat{i}+\hat{j}$ and $2\hat{i}+2\hat{j}$ are
    (a) perpendicular (b) collinear (c) unit vectors (d) equal
14. Midpoint of A(2, 4, 6), B(0, 2, 0)
    (a) $2\hat{i}+6\hat{j}+6\hat{k}$ (b) $\hat{i}+\hat{j}+3\hat{k}$ (c) $\hat{i}+3\hat{j}+3\hat{k}$ (d) $2\hat{i}+2\hat{j}+6\hat{k}$
15. $\vec{a}\times \vec{a}=$
    (a) $|\vec{a}{|}^2$ (b) $\vec{a}$ (c) 1 (d) $\vec{0}$
16. Angle between $\hat{i}$ and $\hat{k}$
    (a) $\frac{\pi }{2}$ (b) 0 (c) $\pi$ (d) $\frac{\pi }{4}$
17. $\vec{a}\cdot \vec{b}=0$ and $\vec{a}\times \vec{b}=\vec{0}$ mean
    (a) $\vec{a}=\vec{0}$ or $\vec{b}=\vec{0}$ (b) $\vec{a}\perp \vec{b}$ (c) $\vec{a}\parallel \vec{b}$ (d) $\vec{a}=\vec{b}$
18. $|\vec{a}+\vec{b}{|}^2=|\vec{a}{|}^2+|\vec{b}{|}^2$ iff
    (a) $\vec{a}\parallel \vec{b}$ (b) $|\vec{a}|=|\vec{b}|$ (c) $\vec{a}=\vec{b}$ (d) $\vec{a}\perp \vec{b}$
19. Scalar components of the vector from (1, 2) to (4, 6)
    (a) 4, 6 (b) 3, 4 (c) 5, 8 (d) 1, 2
20. $\hat{j}\cdot (\hat{k}\times \hat{i})=$
    (a) 0 (b) –1 (c) 1 (d) 3

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: $|\hat{i}+\hat{j}+\hat{k}|=\sqrt{3}$. R: $|x\hat{i}+y\hat{j}+z\hat{k}|=\sqrt{x^2+y^2+z^2}$.
2. A: $2\hat{i}-3\hat{j}+4\hat{k}$ and $-4\hat{i}+6\hat{j}-8\hat{k}$ are collinear. R: Their components are proportional.
3. A: $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$. R: $\vec{a}\times \vec{b}=\vec{b}\times \vec{a}$.
4. A: The projection of $\hat{i}$ on $\hat{j}$ is 1. R: The projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$.
5. A: The area of the triangle with sides $\vec{a},\vec{b}$ is $\frac{1}{2}|\vec{a}\times \vec{b}|$. R: $|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$.
6. A: For unit vectors at angle $\frac{2\pi }{3}$, $\vec{a}+\vec{b}$ is a unit vector. R: $\vec{a}\cdot \vec{b}=0$ for perpendicular vectors.

Key: 1 (a) · 2 (a) · 3 (c) · 4 (d) · 5 (a) · 6 (b)

Answers to the practice set