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Vector Algebra

Multiple Choice Questions

521.

If vectors $\hat{i} + \hat{j} + \hat{k}, \hat{i} - \hat{j} + \hat{k} and 2 \hat{i} + 3 \hat{j} + λ \hat{k}$ are coplanar, then $λ$ is equal to

522.

Given, $\vec{a} ⊥ \vec{b}$ , $|\vec{a}| = 1$ and if $(\vec{a} + 3 \vec{b}) (2 \vec{a} - \vec{b}) = - 10$ , $|\vec{b}|$ is equal to

523.

$[\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = [\vec{a} \vec{b} \vec{c}]$ , then

$[\vec{a} \vec{b} \vec{c}] = 1$
$\vec{a} \vec{b} \vec{c}$ are coplanar
$[\vec{a} \vec{b} \vec{c}] = - 1$
$\vec{a} \vec{b} \vec{c}$ are mutually perpendicular

524.
Area of rhombus is ..., where diagonals are $\vec{a} = 2 \hat{i} - 3 \hat{j} + 5 \hat{k} and \vec{b} = - \hat{i} + \hat{j} + \hat{k}$

$\sqrt{21.5}$

$\sqrt{31.5}$

$\sqrt{28.5}$

$\sqrt{38.5}$

$\sqrt{28.5}$

$Given diagonals of a rhombus are \vec{a} = 2 \hat{i} - 3 \hat{j} + 5 \hat{k} and \vec{b} = - \hat{i} + \hat{j} + \hat{k} Area of rhombus = \frac{1}{2} |\vec{a} \times \vec{b}| = \frac{1}{2} |(2 \hat{i} - 3 \hat{j} + 5 \hat{k}) \times (- \hat{i} + \hat{j} + \hat{k})| = \frac{1}{2} |- 8 \hat{i} - 7 \hat{j} - \hat{k}| = \frac{1}{2} \sqrt{114} = \sqrt{28.5}$

525.

$\frac{\vec{a} . (\vec{b} \times \vec{c})}{\vec{b} . (\vec{c} \times \vec{a})} + \frac{\vec{b} . (\vec{a} \times \vec{b})}{\vec{a} . (\vec{b} \times \vec{c})}$ is equal to

1
2
0
$\infty$

526.

If $|\vec{a}| = |\vec{b}| = 1 and |\vec{a} + \vec{b}| = \sqrt{3}$ , then the value of $(3 \vec{a} - 4 \vec{b}) . (2 \vec{a} + 5 \vec{b})$ is

- 21
$- \frac{21}{2}$
21
$\frac{21}{2}$

527.

If $\vec{a}$ is perpendicular to $\vec{b} and \vec{c}$ , $|\vec{a}|$ = 2, $|\vec{b}|$ = 3, $|\vec{c}|$ = 4 and the angle between $\vec{b} and \vec{c}$ is $\frac{2 π}{3}$ , then [ $\vec{a} \vec{b} \vec{c}$ ] is equal to

$4 \sqrt{3}$
$6 \sqrt{3}$
$12 \sqrt{3}$
$18 \sqrt{3}$

528.

If $\vec{a}$ , $\vec{b} and \vec{c}$ are perpendicular to $\vec{b} + \vec{c}$ , $\vec{c} + \vec{a}$ and $\vec{a} + \vec{b}$ respectively and if $|\vec{a} + \vec{b}|$ = 6, $|\vec{b} + \vec{c}|$ = 8 and $|\vec{c} + \vec{a}|$ = 10 then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

$5 \sqrt{2}$
50
$10 \sqrt{2}$
10

529.

If the vectors $\vec{a} + λ \vec{b} + 3 \vec{c}$ , $- 2 \vec{a} + 3 \vec{b} - 4 \vec{c}$ and $\vec{a} - 3 \vec{b} + 5 \vec{c}$ are coplanar, then the value of $λ$ is