Question

Given that a, b, c are not all zero, find parametric equations for a line in R^3 that passes through the point (x0,y0,z0) and is perpendicular to the line x=x0+at, y=y0+bt, z= z0+ct

Answer 1

We have the direction of the line given represented by the vector <a,b,c>

We need a vector perpendicular to that, <d,e,f>

For two vectors to be perpendicular we want the dot product to be 0
<a,b,c>.<d,e,f>=0
ad+be+cf=0

by observation one choice is d=bc,e=ac,f=-2ba. We can check this

abc+bac-2cba=2abc-2abc=0

now we have the proper direction, and we know it passes through the point (x0,y0,z0)

so we can parametrize it as following:

x=x0+(bc)t
y=y0+(ac)t
z=z0+(-2ba)t