Question

If two adjacent sides of a triangle are represented by vectors $\hat{i}+2\hat{j}+2\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k},$ then the area of triangle.

• A. $\frac{\sqrt{66}}{2}$
• B. $\frac{\sqrt{116}}{3}$
• C. $\frac{\sqrt{56}}{2}$
• D. $\frac{\sqrt{116}}{2}$

Answer 1

Answer: (D)

$\frac{\sqrt{116}}{2}$

Given:-

$\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k}$

$\overrightarrow{b}=3\hat{i}-2\hat{j}+\hat{k}$

Area of triangle is calculated with the Cross product of 2 vectors

$Area=\frac{1}{2}\left|\left(\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 2\\ 3& -2& 1\end{array}\right)\right|$

$=\frac{1}{2}(\hat{i}\left(6\right)-\hat{j}(-4)+\hat{k}(-2-6))$

$=\frac{1}{2}\left|(6\hat{i}+4\hat{j}-8\hat{k})\right|$

$=\frac{1}{2}\sqrt{6^2+4^2+8^2}=\frac{\sqrt{116}}{2}$