### 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 �P 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| 