Question

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

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

Answer 1

Answer: (D)

$\overrightarrow{A B}=(1,1,1), \overrightarrow{A C}=(1,3,5)$
$\Rightarrow \overrightarrow{B C}=\overrightarrow{A C}-\overrightarrow{A B}=(0,2,4)$
Since $D$ is the mid-point of $\overrightarrow{B C}$,
$ \therefore \overrightarrow{B D}=(0,1,2)$
$\overrightarrow{A D}=\overrightarrow{A B}+\overrightarrow{B D}$
$=(1,2,3)$
$\therefore |\overrightarrow{A D}| =\sqrt{1+4+9} $
$=\sqrt{14}$