Question

The two vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}+5\hat{k}$ represent the two sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ respectively of a triangle $ABC.$ The length of the median through $A$ is

• A. $7$
• B. $14$
• C. $\sqrt{14}$
• D. $\frac{\sqrt{14}}{2}$

Answer 1

The correct option is C $\sqrt{14}$


$\overrightarrow{AB}=\hat{i}+\hat{j}+\hat{k}$
$\overrightarrow{AC}=\hat{i}+3\hat{j}+5\hat{k}$
$\Rightarrow \overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB}=2\hat{j}+4\hat{k}$
Since $D$ is the mid-point of $\overrightarrow{BC},$
$\overrightarrow{BD}=\frac{\overrightarrow{BC}}{2}=\hat{j}+2\hat{k}$

Now, in $△ABD,$
$\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$
$\therefore |\overrightarrow{AD}|=\sqrt{1^2+2^2+3^2}=\sqrt{14}$