JEE Main 2023 (Online) 6th April Morning ShiftQuestion 1
Let the image of the point $P(1,2,3)$ in the plane $2x-y+z=9$ be $Q$. If the coordinates of the point $R$ are $(6,10,7)$, then the square of the area of the triangle $PQR$ is $\_\_\_\_\_\_$ .
3D Geometry
Answer$594$
Solution
$2-2+3-9=-6$, $|n|^2=6$, so $Q=P+2(2,-1,1)=(5,0,5)$. $\vec{PQ}=(4,-2,2)$, $\vec{PR}=(5,8,4)$, cross $=(-24,-6,42)$, $|\cdot|^2=2376$. Area$^2=\frac{2376}4=594$.