Question

Prove that the line joining points $A\equiv (7,8,6),B\equiv (9,11,7)$ is $\perp$ to the line joining $C\equiv (9,1,-6)$ and $D\equiv (7,2,-5)$.

Answer 1

$A\equiv (7,8,6)$

$B\equiv (9,11,7)$

$C\equiv (9,1,-6)$

$D\equiv (7,2,-5)$

$AB=(2,3,1)$

$CD=(-2,1,1)$

If the lines are perpendicular then their dot product will equal $0$.

Required to prove $AB\cdot CD=0$

LHS $=(2i+3j+k)\cdot (-2i+j+k)$

$=[2\times (-2\left)\right]+[3\times 1]+[1\times 1]$

$=-4+3+1$

$=0$

RHS$=0$

LHS$=$RHS

Thus prove that the line joining points A & B are perpendicular to the line joining C & D.