Answer to Question #146184 in Analytic Geometry for Dolly

Question

Q.No.1 Use vectors to decide whether the triangle with vertices P(1,-3,-2), Q(2,0,-4) and R(6,-2,-5) is right-angled.

Answer 1

Let us find the coordinates of the vectors $\overrightarrow{QP}$ and $\overrightarrow{QR}:$

$\overrightarrow{QP}=(1-2,-3-0,-2-(-4))=(-1,-3,2)$

$\overrightarrow{QR}=(6-2,-2-0,-5-(-4))=(4,-2,-1)$

Since the scalar product $\overrightarrow{QP}\cdot\overrightarrow{QR}=4(-1)-3(-2)+2(-1)=-4+6-2=0$ , we conclude that $\angle PQR=90^\circ$, and therefore the triangle with vertices $P(1,-3,-2), Q(2,0,-4)$ and $R(6,-2,-5)$ is right-angled.