Question

How to find a point on a line in 3d given the distance from one of the points?

Answer 1

Let a line in three dimensions be specified by two points  and  lying on it, so a vector along the line is given by

(1)The squared distance between a point on the line with parameter  and a point  is therefore(2)

To minimize the distance, set  and solve for  to obtain

(3)

where  denotes the dot product. The minimum distance can then be found by plugging  back into (2) to obtain

(4)(5)