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Vector Algebra

Multiple Choice Questions

301.

$(a . \hat{i}) \hat{i} + (a . \hat{j}) \hat{j} + (a . \hat{k}) \hat{k}$ is equal to

302.

If $|a| = 3, |b| = 4$ , then a value of $λ$ for which a + $λb$ is perpendicular to a - $λb$ is

$\frac{9}{16}$
$\frac{3}{4}$
$\frac{3}{2}$
$\frac{4}{3}$

303.

The projection of $a = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}$ on $b = \hat{i} + 2 \hat{j} + 3 \hat{k}$ is

$\frac{1}{\sqrt{14}}$
$\frac{2}{\sqrt{14}}$
$\sqrt{14}$
$\frac{- 2}{\sqrt{14}}$

304.

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 is equal to

0
38
- 10
10

305.

If $|a \times b| = 4 and |a . b| = 2, then {|a|}^{2} {|b|}^{2}$ is equal to

306.

The angle between the vectors a + b and a - b when a = (1, 1, 4) and b = (1, - 1, 4) is

45°
90°
15°
30°

307.

If a, b, c are mutually perpendicular unit vectors, then $|a + b + c|$ is equal to

3
$\sqrt{3}$
0
1

308.

If ABCDEF is a regular hexagon, then AD + EB + FC equals

0
2 AB
4 AB
3 AB

309.

The projection of the vector $2 \hat{i} + \hat{j} - 3 \hat{k}$ on the vector $\hat{i} - 2 \hat{j} + \hat{k}$ is

$- \frac{3}{\sqrt{14}}$
$\frac{3}{\sqrt{14}}$
$- \sqrt{\frac{3}{2}}$
$\frac{3}{\sqrt{2}}$

310.
An unit vector perpendicular to the plane containing the vectors $\hat{i} - \hat{j} + \hat{k}$ and $- \hat{i} + \hat{j} + \hat{k}$ is

$\pm \frac{\hat{i} - \hat{j}}{\sqrt{2}}$

$\frac{\hat{i} + \hat{k}}{\sqrt{2}}$

$\pm \frac{\hat{j} - \hat{k}}{\sqrt{2}}$

$\mp \frac{\hat{i} + \hat{j}}{\sqrt{2}}$

$\mp \frac{\hat{i} + \hat{j}}{\sqrt{2}}$

$Let a = \hat{i} - \hat{j} + \hat{k} and b = - \hat{i} + \hat{j} + \hat{k} And unit vector perpendicular to the plane a and b = \pm \frac{a \times b}{|a \times b|} Now, a \times b = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 1 & 1 \\ - 1 & 1 & 1 \end{matrix}| = \hat{i} (- 1 - 1) - \hat{j} (1 + 1) + \hat{k} (1 - 1) = - 2 \hat{i} - 2 \hat{j} |a \times b| = \sqrt{2^{2} + 2^{2}} = 2 \sqrt{2} ∴ \frac{a \times b}{|a \times b|} = \mp \frac{2 (\hat{i} + \hat{j})}{2 \sqrt{2}} = \mp \frac{\hat{i} + \hat{j}}{\sqrt{2}}$

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Multiple Choice Questions

310. An unit vector perpendicular to the plane containing the vectors i^ - j^ + k^ and - i^ + j^ + k^ is ± i^ - j^2 i^ + k^2 ± j^ - k^2 ∓ i^ + j^2