If \(\mathbf{P}=3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{Q}=\hat{\mathbf{i}}+2…
If \(\mathbf{P}=3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{Q}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are two sides of a triangle, then its area is equal to ____ sq units.
\(\frac{\sqrt{390}}{4}\)
\(\sqrt{390}\)
\(\frac{\sqrt{390}}{2}\)
\(\frac{\sqrt{390}}{8}\)
Solution
Sides of a triangle given as
\(\begin{aligned}
& \mathbf{P}=3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}} \text { and } \mathbf{Q}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \\
& \therefore \text { Area of triangle }=\frac{1}{2}(\mathbf{P} \times \mathbf{Q}) \\
& \left.=\frac{1}{2}|| \begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
3 & 5 & -1 \\
1 & 2 & 3
\end{array} \right\rvert\, \\
& =\frac{1}{2}|\hat{\mathbf{i}}(15+2)-\hat{\mathbf{j}}(9+1)+\hat{\mathbf{k}}(6-5)| \\
& =\frac{1}{2}|17 \hat{\mathbf{i}}-10 \hat{\mathbf{j}}+\hat{\mathbf{k}}|=\frac{1}{2} \sqrt{289+100+1} \\
& =\frac{1}{2} \sqrt{390} \text { sq. units } \\
\end{aligned}\)
Hence, option (c) is correct.