Difficult vector question

Feb 2008
297
7
Shortest distance between point and line, dot product

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

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

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

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

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

\(\displaystyle
\vec{PB}=\mathbf{b} - \mathbf{a} - proj_{\mathbf{d}}(\mathbf{b} - \mathbf{a})\), and show that \(\displaystyle \vec{PB}\) is orthogonal to the direction \(\displaystyle \mathbf{d}\) of the line
 
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