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.

Question

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., $\frac{\sqrt{390}}{4}$, $\sqrt{390}$, $\frac{\sqrt{390}}{2}$, $\frac{\sqrt{390}}{8}$

MARKS App · Accepted Answer

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.

Search any question & find its solution

Looking for more such questions to practice?