Find the distance the point P(10,5,2) is to the line through the two points Q(5,2,2) and R(8,1,1).

Answer 1

The idea is that the area of the parallelogram spanned by QP and QR depends on the distance between P and QR. We will find the parallelogram area with a vector product and then solve for the distance.

Form vectors from one point to another:

QP = P - Q = (10,5,2) - (5,2,2) = (5,3,0)

QR = R - Q = (8,1,1) - (5,2,2) = (3,-1,-1)

Find their vector product with any method from your textbook:

QP x QR = (5,3,0) x (3,-1,-1) = (-3,5,-14)

The area of the parallelogram spanned by QP and QR is the length of the vector product:

area = length of (-3,5,-14) = root(3² + 5² + 14²) = √230

This area equals the length of the base QR = root(3² + 1² + 1²) =√11, times the perpendicular distance (height) from point P to the base QR:

area = base . height

So the perpendicular distance (height) that you want is:

height = area/base = root(230/11)