So the distance from 𝑃=(5,−4,4) P = ( 5 , − 4 , 4 ) to the line through the points 𝐴=(2,5,1) A = ( 2 , 5 , 1 ) and 𝐵=(5,4,−2) B = ( 5 , 4 , − 2 ) is

Question

the distance d of a point P to the line through the points A and B is the length of the component of AP that is orthogonal to AB

So the distance from 𝑃=(5,−4,4) P = ( 5 , − 4 , 4 ) to the line through the points 𝐴=(2,5,1) A = ( 2 , 5 , 1 ) and 𝐵=(5,4,−2) B = ( 5 , 4 , − 2 ) is

Answer 1

The directing vector of line AB is u= < 3, -1 -3 >

Vector AP = < 3, -9, 3 >

Then the distance of the point P from the line AB is equal with the

length of the projection of vector AP on vector u.

| AP • u | / || u|| = | < 3, -9, 3 > • < 3, -1 -3 > | / √19 = 9√19 /19