First slide

Vectors

Question

Two adjacent sides of a parallelogram are represented by the two vectors $\hat{i} + 2 \hat{j} + 3 \hat{k} and 3 \hat{i} - 2 \hat{j} + \hat{k}$ . What is the area of parallelogram

Moderate

A

8
B

$8 \sqrt{3}$
C

$3 \sqrt{8}$
D

192

Solution

Area of parallelogram = $\vec{A} \times \vec{B}$
$= (\hat{i} + 2 \hat{j} + 3 \hat{k}) \times (3 \hat{i} - 2 \hat{j} + \hat{k})$
= $(\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 3 & - 2 & 1 \end{matrix})$ = $(8) \hat{i} + (8) \hat{j} - (8) \hat{k}$
Magnitude = $\sqrt{64 + 64 + 64} = 8 \sqrt{3}$

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