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Thread: Difficult vector question

  1. #1
    Senior Member
    Feb 2008

    Shortest distance between point and line, dot product

    Let B be a point \mathbb{R}^n with coordinate vector \mathbf{b}. Let \mathbf{x=a+\lambda d}, \lambda \in \mathbb{R} be the equation of a line.

    1) Show that the square of the distance from B to an arbritrary point \mathbf{x} on the line is given by:

    q(\lambda)=|\mathbf{b}-\mathbf{a}|^2-2\lambda(\mathbf{b}-\mathbf{a}) \cdot \mathbf{d}+ \lambda^2 |\mathbf{d}|^2

    2) Find the shortest distance between the point B and the line by minimising q(\lambda)

    3) If P is the point on the line closest to B, show that:

     <br />
\vec{PB}=\mathbf{b} - \mathbf{a} - proj_{\mathbf{d}}(\mathbf{b} - \mathbf{a}), and show that \vec{PB} is orthogonal to the direction \mathbf{d} of the line
    Last edited by acevipa; Jun 1st 2010 at 07:25 AM.
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