Point-Line Distance--3-Dimensional

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)
	(6)

Using the vector quadruple product

(7)

where denotes the cross product then gives

(8)

and taking the square root results in the beautiful formula

			(9)
			(10)
			(11)

Here, the numerator is simply twice the area of the triangle formed by points , , and , and the denominator is the length of one of the bases of the triangle, which follows since, from the usual triangle area formula, .