Vector-01

Comprehensive Vector Math Quiz

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1. If $\vec{a}$ lies in the plane of vectors $\vec{b}$ and $\vec{c}$, then which of the following is correct?

(a) $[\vec{a} \vec{b} \vec{c}] = 0$ (b) $[\vec{a} \vec{b} \vec{c}] = 1$ (c) $[\vec{a} \vec{b} \vec{c}] = 3$ (d) $[\vec{b} \vec{c} \vec{a}] = 1$

2. The value of $[\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}]$, where $|\vec{a}|=1, |\vec{b}|=5, |\vec{c}|=3$, is

(a) 0 (b) 1 (c) 6 (d) none of these

3. If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-coplanar mutually perpendicular unit vectors, then $[\vec{a} \vec{b} \vec{c}]^2$ is

(a) $\pm 1$ (b) 0 (c) $-2$ (d) 1

4. If $\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0$ for some non-zero vector $\vec{r}$, then the value of $[\vec{a} \vec{b} \vec{c}]$, is

(a) 2 (b) 3 (c) 0 (d) none of these

5. For any three vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ the expression $(\vec{a} - \vec{b}) \cdot \{ (\vec{b} - \vec{c}) \times (\vec{c} - \vec{a}) \}$ equals

(a) $[\vec{a} \vec{b} \vec{c}]$ (b) $2[\vec{a} \vec{b} \vec{c}]$ (c) $[\vec{a} \vec{b} \vec{c}]^2$ (d) none of these

6. If $\vec{a}$, $\vec{b}$, $\vec{c}$ are non-coplanar vectors, then $\frac{(\vec{a} \times \vec{b}) \cdot \vec{c}}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{(\vec{b} \times \vec{c}) \cdot \vec{a}}{(\vec{c} \times \vec{b}) \cdot \vec{a}}$ is equal to

(a) 0 (b) 2 (c) 1 (d) none of these

7. Let $\vec{a}_1 = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, $\vec{b}_1 = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ and $\vec{c}_1 = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ be three non-zero vectors such that $\vec{c}_1$ is a unit vector perpendicular to both $\vec{a}_1$ and $\vec{b}_1$. If the angle between $\vec{a}_1$ and $\vec{b}_1$ is $\frac{\pi}{6}$, then $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}^2$ is equal to

(a) 0 (b) 1 (c) $\frac{1}{4}|\vec{a}|^2 |\vec{b}|^2$ (d) $\frac{3}{4}|\vec{a}|^2 |\vec{b}|^2$

8. If $\vec{a} = 2\hat{i} - 3\hat{j} + 5\hat{k}$, $\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$ and $\vec{c} = 5\hat{i} - 3\hat{j} - 2\hat{k}$, then the volume of the parallelepiped with conterminous edges $\vec{a} + \vec{b}$, $\vec{b} + \vec{c}$, $\vec{c} + \vec{a}$ is

(a) 2 (b) 1 (c) $-1$ (d) 0

9. If $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{b} \times \vec{c} = \vec{a}$, then $|\vec{b}|$ is equal to (Based on common problems, actual Q9 in image is cut off)

(a) 2 (b) 1 (c) $\frac{1}{\sqrt{2}}$ (d) $\frac{1}{2}$

10. The value of $(\vec{a} \times \vec{b})^2 + (\vec{a} \cdot \vec{b})^2$ is

(a) $|\vec{a}|^2 |\vec{b}|^2$ (b) $|\vec{a}|^2 + |\vec{b}|^2$ (c) $|\vec{a}|^2 + |\vec{b}|^2 |\vec{a}|^2$ (d) $2|\vec{a}|^2 |\vec{b}|^2$

11. If the vectors $4\hat{i} + 11\hat{j} + m\hat{k}$, $7\hat{i} + 2\hat{j} + 6\hat{k}$ and $\hat{i} + 5\hat{j} + 4\hat{k}$ are coplanar, then $m = $

(a) 0 (b) 38 (c) $-10$ (d) 10

12. For non-zero vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ the relation $|(\vec{a} \times \vec{b}) \cdot \vec{c}| = |\vec{a}| |\vec{b}| |\vec{c}|$ holds good, if

(a) $\vec{a} = \vec{b} = \vec{c} = 0$ (b) $\vec{a} \cdot \vec{b} = 0, \vec{a} \cdot \vec{c} = 0$ (approx) (c) $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$ (d) $\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$ (approx)

13. If $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$, then $(\vec{a} + \vec{b}) \cdot (\vec{b} + \vec{c}) \times (\vec{c} + \vec{a})$ is

(a) 0 (b) $[\vec{a} \vec{b} \vec{c}]$ (c) $2[\vec{a} \vec{b} \vec{c}]$ (d) $[\vec{a} \vec{b} \vec{c}]$ (repeated from b)

14. If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-coplanar vectors, then $[(\vec{a} + \vec{b}) (\vec{b} + \vec{c}) (\vec{c} + \vec{a})]$ equals

(a) 0 (b) $[\vec{a} \vec{b} \vec{c}]$ (c) $2[\vec{a} \vec{b} \vec{c}]$ (d) $-[\vec{a} \vec{b} \vec{c}]$

15. $(\vec{a} + 2\vec{b} - \vec{c}) \cdot ((\vec{a} - \vec{b}) \times (\vec{a} - \vec{b} - \vec{c}))$ is equal to

(a) $[\vec{a} \vec{b} \vec{c}]$ (b) $2[\vec{a} \vec{b} \vec{c}]$ (c) 0 (d) $[\vec{a} \vec{b} \vec{c}]$ (repeated from a)

16. The vectors $\hat{i} + \hat{j} + 2\hat{k}$, $\hat{i} + \lambda\hat{j} - \hat{k}$ and $2\hat{i} - \hat{j} + \lambda\hat{k}$ are coplanar, if $\lambda = $

(a) $-2$ (b) 0 (c) 1 (d) $-1$

17. The vectors $3\hat{i} - \hat{j} + 2\hat{k}$, $\hat{i} + 3\hat{j} - \hat{k}$ and $4\hat{i} + \lambda\hat{j} - \hat{k}$ are coplanar, if the value of $\lambda$ is

(a) $-2$ (b) 0 (c) 2 (d) any real number

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