Distance from a point to a line

The problem

Let

, and

be the position vectors of the points A, B and C respectively,

and L be the line passing through A and B.

Find the shortest distance from C to L.

Method 1 By Pythagoras Theorem

The vector equation of the line, L, which passes through A and B:

A unit vector along AB :

Let D be the foot of the perpendicular from C to L. Then

and |AC| = Ö14

\ The shortest distance from C to AB = CD

Method 2 Using Cross Product

Consider the cross product:

Remember the magnitude of this cross product gives the area of the parallelogram with sides given by the vector AC and AB.

and the side of AB is given by:

Therefore the height of the parallelogram, which gives the distance of C to AB

Method 3 Using Dot Product

The vector equation of the line, L, which passes through A and B:

Therefore a point D on L is given by:

Now, since CD is perpendicular to AB, CD · AB = 0

\ 2(2t + 3) + 3(3t - 1) + 1(t+2) = 0

The required distance

Method 4 Using the Concept of distance from a point to a line

As in method 3, we find

The distance of a point to the line is the minimum of all distances from that point to any point on the line.

\ The required distance = minimum of |CD|