find the point of the line vector x =... that is closest to (1, -3, 0)

Question

Find the point on the line vector x = (1, -1, -1) + t(1, 1, 1) that is closest to (1, -3, 0)

Answer 1

there are 2 ways to approach this. a more straight forward way is to find the generic distance formula between any given point on the line and (1,-3,0), differentiate and set to 0 (to find the minimum distance, hence the closest point)

or you can use the dot product of a vector from line to the point and a vector along the line set to 0 as well.

the answer is (4/3,-2/3,-2/3)