Question

Find parametric equations for the line passing through the point P(4,5,5), intersecting the line <x, y, z> = <11, -8, 4> + t <3, -1,1> at a 90 degree angle.

Answer 1

Let P be the point (4,5,5) and L be the line given by x= 11+3t, y=-8-t, z=4+t

We know that the new line intersects L at some point, so let this point be Q=(11+3d, -8-d, 4+d). Since these line intersect at 90 degree, if we take the dot product of the vector from P to Q and a vector parallel to L, we should get 0. With this information we can solve for d.

We have  

0= (7+3d, -13-d,-1+d) . (3,-1,1)

= 21+9d+13+d-1+d

= 33+11d

Hence d=-3

Now we have two points on the line we are looking for:

P=(4,5,5) and Q=(2,-11,1) and this is enough to determine a valid parameterization (we let t=0 correspond to P and t=1 correspond to Q):

x=4-2t, y=5-16t, z=5-4t