Question

The adjacent sides $AB$ and $AC$ of a triangle $ABC$ are represented by the vectors $-2\hat{i}+3\hat{j}+2\hat{k}$ and $-4\hat{i}+5\hat{j}+2\hat{k}$ respectively. The area of the triangle $ABC$ is.

• A. $6$ square units
• B. $5$ square units
• C. $4$ square units
• D. $3$ square units

Answer 1

The correct option is D $3$ square units

Area of triangle is given as $\frac{1}{2}|\overrightarrow{a}||\overrightarrow{b}\sin \theta =\frac{1}{2}\sqrt{17\times 45}\sin \theta$

Also, $\cos \theta =\frac{8+15+4}{\sqrt{17\times 45}}$

$\therefore \sin \theta =\sqrt{\frac{17\times 45-{27}^2}{17\times 45}}=\frac{6}{\sqrt{17\times 45}}$

$\therefore$ area $=\frac{1}{2}\sqrt{17\times 45}\times \frac{6}{\sqrt{17\times 45}}=3$ sq. units