First slide

Scalars and Vectors

Question

The area of the triangle formed by $2 \hat{i} + \hat{j} - \hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$ is_____

Moderate

A

3 sq. unit
B

$2 \sqrt{3}$ sq. unit
C

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

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

Solution

Area $= \frac{1}{2} | \vec{A} \times \vec{B} |$
$\vec{A} \times \vec{B} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & - 1 \\ 1 & 1 & 1 \end{matrix} |$

$= \hat{i} (1 + 1) + \hat{j} (- 1 - 2) + \hat{k} (2 - 1) = 2 \hat{i} - 3 \hat{j} + \hat{k}$

$∴$ Area $= \frac{1}{2} \sqrt{4 + 9 + 1} = \frac{\sqrt{14}}{2} sq . units$

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