Question

If the position vectors of $A,B,C,D$ are respectively $(-6,1,6)$ $(6,-2,3)$, $(-2,-3,-1)$ and $(-5,-9,-7)$, then

• A. $\angle BCA$ is a right angle
• B. $\angle CDA$ is a right angle
• C. $\angle BDA$ is a right angle
• D. $\angle ACD$ is a right angle

Answer 1

Answer: (A)

$\angle BCA$ is a right angle

Looking at the given position vectors, we get $\overline{BC}=-8\overline{i}-1\overline{j}-4\overline{k}$
$\overline{AC}=4\overline{i}-4\overline{j}-7\overline{k}$
$\overline{CD}=-3\overline{i}-6\overline{j}-6\overline{k}$
$\overline{AD}=1\overline{i}-10\overline{j}-13\overline{k}$
$\overline{AB}=12\overline{i}-3\overline{j}-3\overline{k}$
$\overline{BD}=-11\overline{i}-7\overline{j}-10\overline{k}$
Thus, $\overline{AC}.\overline{BC}=0$ and hence option $A$ is correct.