Question

Show that the line through the points $(1,-1,2),(3,4,-2)$ is perpendicular to the line through the points $(0,3,2)$ and $(3,5,6)$.

Answer 1

Given $A(1,-1,2),B(3,4,-2),C(0,3,2),D(3,5,6)$

The dr's of line joining $A$ and $B$ is

$(x_2-x_1),(y_2-y_1),(z_2-z_1)=(3-1,4+1,-2-2)=(2,5,-4)$

The dr's of line joining $C$ and $D$ is

$(x_2-x_1),(y_2-y_1),(z_2-z_1)=(3-0,5-3,6-2)=(3,2,4)$

For the lines $AB$ and $CD$ to be perpendicular

$a_1{a}_2+b_1{b}_2+c_1{c}_2=0$ ($a_1=2,b_1=5,c_1=-4,A_2=3,b_2=2,c_2=4$)

$2\left(3\right)+5\left(2\right)+(-4)\left(4\right)$

$=6+10-16=16-16=0$

$\therefore$ $AB\perp CD$