Question

If $A(1,2,3),B(2,3,1),C(3,1,2)$ are vertices of $△ABC$, then the length of altitude through $C$ is

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

Answer 1

The correct option is D $\frac{3}{\sqrt{2}}$

Given: $A(1,2,3),B(2,3,1),C(3,1,2)$
We know, the length of altitude through $C=\frac{|\overrightarrow{CA}\times \overrightarrow{CB}|}{|\overrightarrow{AB}|}\cdots \left(i\right)$
$\overrightarrow{CA}=-2\hat{i}+\hat{j}+\hat{k}\overrightarrow{CB}=-\hat{i}+2\hat{j}-\hat{k}\overrightarrow{AB}=\hat{i}+\hat{j}-2\hat{k}\Rightarrow |\overrightarrow{CA}\times \overrightarrow{CB}|=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -2& 1& 1\\ -1& 2& -1\end{array}\right|=|-3\hat{i}-3\hat{j}-3\hat{k}|=3\sqrt{3}$
$|\overrightarrow{AB}|=\sqrt{6}$
Hence putting this in $\left(i\right)$, we have required altitude $=\frac{3}{\sqrt{2}}$