Question

Consider $A,B,C,D$ with position vector $7\hat{i}-4\hat{j}+7\hat{k}$, $\hat{i}-6\hat{j}+10\hat{k}$, $-\hat{i}-3\hat{j}+4\hat{k}$ and $5\hat{i}-\hat{j}+5\hat{k}$ respectively, then $ABCD$ is a

• A. Rhombus
• B. Rectangle
• C. Parallelogram but not rhombus
• D. None of the above

Answer 1

Answer: (D)

None of the above

$\overline{AB}=\overline{OB}-\overline{OA}=6\hat{i}+2\hat{j}-3\hat{k}$
$\therefore \left|\overline{AB}\right|=\sqrt{49}=7$

Similarly $\overline{BC}=2\hat{i}-3\hat{j}+6\hat{k}$

$\therefore \left|\overline{BC}\right|=\sqrt{49}=7$
$\overline{CD}=6\hat{i}-2\hat{j}+\hat{k}$

$\therefore \left|\overline{CD}\right|=\sqrt{41}$ $\overline{DA}=-2\hat{i}+3\hat{j}-2\hat{k}$
$\therefore \left|\overline{DA}\right|=\sqrt{17}$

Therefore, correct option is D.