Find the distance from the point (2, 4, 4) to the line x = 0, y = 4 + 3t, z = 4 + 2t.
The cross product and the dot product and d = |n * b|/|n|
The Attempt at a Solution
So the distance from the point to the line is the line directly perpendicular from the point to the line. To find that distance we can find the dot product (which produces a scalar projection of the distance from the point to the line onto the line's normal vector) of the vector from any point on the line (call it q) and the point (call it p) and the line itself. To get the line from p to q (call it pq) I simply do the following:
Set t = 3 and retrieve the following points <0,13,10> and we subtract that from p to get pq = <2,-9,-6>.
Next I need to find the normal vector, which is where I am having trouble with. I can find the direction numbers of the line which results in <0, 3, 2>, however my problem now is that I don't know what I should cross product it with to get the normal vector. If I cross product it with pq I'll get a line coming straight out of the board and for me it seems like that a normal vector that isn't pointing in the same direction as p will screw up the projection. Can you please explain?