## 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 Let D be the
foot of the perpendicular from C to L. Then
\
The shortest distance from C to AB =
CD |

Method 2 Using Cross Product Consider the cross
product: Remember the and the side of 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 \ 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 \ The required distance =
minimum of | |