Question

Show that the lines $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ are perpendicular to each other.

Answer 1

The direction vector of $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}$ is $(7,-5,1)$ and direction vector of $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$

is $(1,2,3)$

Dot product of the vectors is $7\times 1+(-5)\left(2\right)+1\times 3=0$

Therefore the lines are perpendicular.