Find k such that two vectors are perpendicular

Question

Find Value of k so that the lines $\frac{x-1}{k}=\frac{y+2}{3}=\frac{z}{2}$ and $\frac{x+2}{k}=\frac{y+7}{k+2}=z-5$ are perpendicular

Answer 1

The direction of the first is given by the vector $(k,3,2)$ and the direction of the second by $(k,k+2,1).$ These vectors are perpendicular if and only if their dot product is zero. That is

$$(k,3,2)\cdot (k,k+2,1)=k^2+3k+8=0.$$ Study the equation and you are done.

Answer 2

Hints:

The line with equation $\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$ is parallel to vector $<a,b,c>$

Two vectors are orthogonal to each other iff their dot product is zero.

Answer 3

since the dot product is $$k^2+3k+8$$ and this product is for no real $k$ zero the vectors are not perpendicular.