First slide

Scalars and Vectors

Question

The area of the parallelogram whose sides are represented by the vectors $\hat{j} + 3 \hat{k}$ and $\hat{i} + 2 \hat{j} - \hat{k}$ is

Moderate

A

$\sqrt{61}$ sq. unit
B

$\sqrt{59}$ sq. unit
C

$2 \sqrt{14}$ sq. unit
D

$\frac{\sqrt{14}}{2}$ sq. unit

Solution

Area $= | \vec{A} \times \vec{B} |$
$\vec{A} \times \vec{B} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & 3 \\ 1 & 2 & - 1 \end{matrix} |$
$= \hat{i} (- 1 - 6) + \hat{j} (3 - 0) + \hat{k} (0 - 1) = - 7 \hat{i} + 3 \hat{j} - \hat{k}$
$∴ | \vec{A} \times \vec{B} | = \sqrt{49 + 9 + 1} = \sqrt{59}$ sq. units

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