First slide

Vectors

Question

The vectors from origin to the points A and B are $\vec{A} = 3 \hat{i} - 6 \hat{j} + 2 \hat{k} and \vec{B} = 2 \hat{i} + \hat{j} - 2 \hat{k}$ respectively. The area of the triangle OAB be

Moderate

A

$\frac{5}{2} \sqrt{17} sq . unit$
B

$\frac{2}{5} \sqrt{17} sq . unit$
C

$\frac{3}{5} \sqrt{17} sq . unit$
D

$\frac{5}{3} \sqrt{17} sq . unit$

Solution

Given $\bar{OA} = \bar{a} = 3 \hat{i} - 6 \hat{j} + 2 \hat{k} and$
$\bar{OB} = \bar{b} = 2 \hat{i} + \hat{j} - 2 \hat{k}$
$∴ (\vec{a} \times \vec{b}) = (\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & - 6 & 2 \\ 2 & 1 & - 2 \end{matrix})$
= $(12 - 2) \hat{i} + (4 + 6) \hat{j} + (3 + 12) \hat{k}$
= $10 \hat{i} + 10 \hat{j} + 15 \hat{k} \Rightarrow | \vec{a} \times \vec{b} |$
= $\sqrt{10^{2} + 10^{2} + 15^{2}}$
= $\sqrt{425} = 5 \sqrt{17}$
Area of $∆ OAB = = \frac{1}{2} | \vec{a} \times \vec{b} | = \frac{5 \sqrt{17}}{2} sq . unit .$

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