Yeah they are skipping stuff in the 'solution'
Would be better and clearer to turn the 2 lines into directional vector form like
(x,y,z)=(3,-1,1)+ μ(2,3,-2) [you can get this by letting all fractions equal μ ]
Likewise for the other line
Then since they are perpendicular their directional vectors have a dot product of 0 which is your red box
