Multiple choice (Vectors)

Choose and write the correct option in the following question.(vectors multiple choice)

1.) The value of the expression $|\vec{a}\times \vec{b}|^2=(\vec{a}.\vec{b})^2$ is

(a) $\vec{a}.\vec{b}$ (b) $|\vec{a}||\vec{b}|$

Answer (c)

2.) The area of the triangle formed by vertices O, A, B where $\vec{OA} = \hat{i}+2\hat{j}+3\hat{k}$ and $\vec{OB}= -3\hat{i}-2\hat{j}+\hat{k}$ is

(a) $3\sqrt{5}$ sq.units (b) $5\sqrt{5}$ sq.units

Answer (a)

3.) If θ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then $\vec{a}.\vec{b} \geq 0$ only when

(a) $0 <\theta<\frac{\pi}{2}$ (b) $0\leq \theta \leq \frac{\pi}{2}$

Answer (b)

4.) If θ is the angle between any two vectors $\vec{a}$ and $\vec{b}$ then $|\vec{a}-\vec{b}|=|\vec{a}\times \vec{b}|$ , where θ is equal to

(a) 0 (b) π/4

Answer (c)

5.) The vector of the direction of the vector $\hat{i}-2\hat{j}+2\hat{k}$ that has magnitude 9 is

(a) $\hat{i}-2\hat{j}+2\hat{k}$

(b) $\frac{\hat{i}-2\hat{j}+2\hat{k}}{3}$

(d) $9(\hat{i}-2\hat{j}+2\hat{k})$

Answer (c)

6.) The position vector of the point which divides the joining of points $2\vec{a}-3\vec{b}$ and $\vec{a}+\vec{b}$ in the ratio 3:1 is

(a) $\frac{3\vec{a}-2\vec{b}}{2}$

(b) $\frac{7\vec{a}-8\vec{b}}{4}$

(d) $\frac{5\vec{a}}{4}$

Answer (d)

7.) The vector having initial and terminal points as (2, 5, 0) and (-3, 7, 4) respectively is

(a) $-\hat{i}+12\hat{j}+4\hat{k}$

(b) $5\hat{i}+2\hat{j}-4\hat{k}$

(d) $\hat{i}+\hat{j}+\hat{k}$

Answer (c)

8.) The angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitude √3 and 4 respectively and $\vec{a}.\vec{b}=2\sqrt{3}$ is

(a) π/6 (b) π/3

Answer (b)

9.) The value of λ such that the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j}+\hat{k}$ and $\vec{b} = \hat{i}+2\hat{j}+3\hat{k}$ are ortogonal is

(a) 0 (b) 1

Answer (d)

10.) The value of λ for which the vectors $3\hat{i}-6\hat{j} +\hat{k}$ and $2\hat{i} -4\hat{j} +\lambda\hat{k}$ are parallel is

(a) 2/3 (b) 3/2

Answer (a)

11.) The vectors from origin to the points A and B are $\vec{a}=2\hat{i}-3\hat{j}+2\hat{k}$ and $\vec{b} = 2\hat{i}+3\hat{j}+\hat{k}$ .respectively then the area of triangle OAB is

(a) 340 (b) √25

Answer (d)

12.) For any vector $\vec{a}$ , the value of $(\vec{a}\times\hat{i})^2+(\vec{a}\times\hat{j})^2+(\vec{a}\times\hat{k})^2$ is equal to

(a) $\vec{a}^2$ (b) $3\vec{a}^2$

Answer (d)

13.) If $|\vec{a}| = 10,|\vec{b}| = 2$ and $\vec{a}.\vec{b} = 12$ , then value of $|\vec{a}\times \vec{b}|$ is

(a) 5 (b) 10

Answer (d)

14.) The projection of vector $\hat{i}-2\hat{j}+\hat{k}$ on the vector $4\hat{i}-4\hat{j} +7\hat{k}$ is

(a) $\frac{5\sqrt{6}}{10}$ (b) $\frac{19}{9}$

Answer (b)

15.) If $\vec{a},\vec{b},\vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c} = 0$ , then the value of $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ is

(a) I (b) 3

Answer (c)

16.) Projection vector of $\vec{a}$ on $\vec{b}$ is

(a) $\left(\frac{|\vec{a}.\vec{b}|}{|\vec{b}|^2}\right)\vec{b}$

(b) $\frac{\vec{a}.\vec{b}}{|\vec{b}|}$

(d) $\left(\frac{\vec{a}.\vec{b}}{|\vec{b}|^2}\right)\vec{b}$

Answer (a)

17.) If $\vec{a},\vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a}+\vec{b}+\vec{c}=0$ and $|\vec{a}|=2,|\vec{b}| = 3, |\vec{c}| = 5$ then value of $\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$ is

(a) 0 (b) 1

Answer (c)

18.) If $|\vec{a}| = 4$ and -3 ≤ λ ≤ 2, then the range of $|\lambda\hat{a}|$ is

(a) [0, 8] (b) [-12, 8]

Answer (c)

19.) The number of vectors of unit length perpendicular to the vectors $\vec{a} = 2\hat{i}+\hat{j}+21\hat{k}$ and $\vec{b} = \hat{j}+\hat{k}$ is

(a) one (b) two

Answer (b)

20.) The position vector of the point which divides the jpin of points with position vectors $\vec{a}+\vec{b}$ and $2\vec{a} - \vec{b}$ in the ratio 1:2 is