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Vector Algebra

Multiple Choice Questions

181.

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
$\frac{21}{2}$

182.

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

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

183.
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

$Given, |\vec{a} + \vec{b}| = 6 \Rightarrow {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + 2 \vec{a} . \vec{b} = 36 . . . (i) Similarly, {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + 2 \vec{b} . \vec{c} = 64 . . . (ii) and {|\vec{c}|}^{2} + {|\vec{a}|}^{2} + 2 \vec{c} . \vec{a} = 100 . . . (iii) On adding Eqs . (i), (ii) and (iii), we get {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = 100 \Rightarrow {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} = 100 . . . (iv) [∵ \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} = 0] \Rightarrow {|\vec{a} + \vec{b} + \vec{c}|}^{2} = {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) \Rightarrow {|\vec{a} + \vec{b} + \vec{c}|}^{2} = 100 [∵ from Eq . (iv)] \Rightarrow |\vec{a} + \vec{b} + \vec{c}| = 10$

184.

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

185.

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ , $|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7$ , then anle between $\vec{a} and \vec{b}$ is

$\frac{π}{6}$
$\frac{2 π}{3}$
$\frac{5 π}{3}$
$\frac{π}{3}$

186.

If the vectors $\vec{a} = \hat{i} + a \hat{j} + a^{2} \hat{k}$ , $\vec{b} = \hat{i} + b \hat{j} + b^{2} \hat{k}$ and $\vec{c} = \hat{i} + c \hat{j} + c^{2} \hat{k}$ are three non-coplanar vectors and $|\begin{matrix} a & a^{2} & 1 + a^{3} \\ b & b^{2} & 1 + b^{3} \\ c & c^{2} & 1 + c^{3} \end{matrix}| = 0$ , then the value of abc is

187.

Let $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + 2 \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} + \hat{j} - 2 \hat{k}$ be three vectors. A vector in the plane of $\vec{b} and \vec{c}$ whose projection on $\vec{a}$ is magnitude $\sqrt{\frac{2}{3}}$ , is

$2 \hat{i} + 3 \hat{j} - 3 \hat{k}$
$2 \hat{i} + 3 \hat{j} + 3 \hat{k}$
$2 \hat{i} - 5 \hat{j} + 5 \hat{k}$
$2 \hat{i} + \hat{j} + 5 \hat{k}$

188.

If the constant forces $2 \hat{i} - 5 \hat{j} + 6 \hat{k}$ and $- \hat{i} + 2 \hat{j} - \hat{k}$ act on a particle due to which it is displaced from a point A (4,- 3, - 2) to a point B (6, 1,- 3), then the work done by the forces is