First slide

Vectors

Question

If $\vec{A} = 3 \hat{i} + \hat{j} + 2 \hat{k} and \vec{B} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k} then value of | \vec{A} \times \vec{B} | will be$

Moderate

A

$8 \sqrt{2}$
B

$8 \sqrt{3}$
C

$8 \sqrt{5}$
D

$5 \sqrt{8}$

Solution

$\vec{A} \times \vec{B} = (\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & 2 \\ 2 & - 2 & 4 \end{matrix})$
$= (1 \times 4 - 2 \times - 2) \hat{i} - (2 \times 2 - 4 \times 3) \hat{j} + (3 \times - 2 - 1 \times 2) \hat{k}$
= $8 \hat{i} - 8 \hat{j} - 8 \hat{k}$
$∴ Magnitude of \vec{A} \times \vec{B} = | \vec{A} \times \vec{B} |$
$= \sqrt{{(8)}^{2} + {(- 8)}^{2} + {(- 8)}^{2}} = 8 \sqrt{3}$

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