Find the value of $a$ for which two lines are perpendicular

Question

Two vectors are perpendicular when their dot product is zero. But how to get the dot product of two lines?

$r_1= (8,-3,1)+s(12,-5,0)$

$r_2= (1,14,3)+t(5,a,b)$

The question asks for the value of $a$ given that the two lines are perpendicular.

Answer 1

The vectors $(12,-5,0)$ and $(5,a,b)$ are the direction vectors of the lines. If the dot product of those two vectors is zero, then the lines are perpendicular.

Hence you have to calculate: $$(12,-5,0)\cdot(5,a,b)=0$$

This gives: $12\cdot5-5\cdot a=0 \Longleftrightarrow 60-5a=0 \Longrightarrow \boxed{a=12}$