Question

The points $P,Q,R$ have regular cartesian coordinates $(1,1,-1),(4,1,2)$ and $(-2,1,2)$ respectively. Which of the following is true for the triangle $PQR$?

• A. All the sides have the same length.
• B. $QR=2PQ$
• C. Angle $PQR$ is a right angle.
• D. The area of the triangle is $18$ square units.

Answer 1

The correct option is C Angle $PQR$ is a right angle.

$\overline{PQ}=(1-4)\overrightarrow{i}+(1-1)\overrightarrow{j}+(-1-2)\overrightarrow{k}\overline{PQ}=-3i-3k$

$\overline{PR}=(1+2)\overrightarrow{i}+(1-1)\overrightarrow{j}+(-1-2)\overrightarrow{}\overline{PR}=3i-3k$

$\overline{PQ}.\overline{PR}=(-3\overrightarrow{i}-3\overrightarrow{k}).(3i-3k)\overline{PQ}.\overline{PR}=-9+9=0$

As dot product between two sides of a triangle is zero, these two sides forms the angle ${90}^0$ between them.

Hence, the answer is option C.