Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $A B C D$. If the diagonal $\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat{j}+3 \hat{k}$, then the area of the parallelogram is equal to :
Solution
<p>$$\begin{aligned}
& \text { Area }=|\overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{BD}}| \\
& =\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
5 & -1 & 7 \\
1 & 2 & 3
\end{array}\right| \\
& =\frac{1}{2}|-17 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}|=\frac{1}{2} \sqrt{474}
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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