4.5 Projections

The figure below show representations \(\overrightarrow {PQ}\) and \(\overrightarrow {PR}\) of two vectors \(\overline {a}\) and \(\overline {b}\) with the same initial point \(P\).

  

If \(S\) is the foot of the perpendicular from \(R\) to the line containing \(\overrightarrow {PQ}\) then the vector with representations \(\overrightarrow {PS}\) is called the vector projection of \(\overline {b}\) onto \(\overline {a}\) and denoted by \(Proj_{\overline {a}}\overline {b}\). It is given by \[Proj_{\overline {a}}\overline {b} = \frac {\overline {a}\cdot \overline {b}}\,{\left |\overline {a}\right |^2} \overline {a} \]

The scalar projection of \(\overline {b}\) onto \(\overline {a}\) (also called the component of \(\overline {b}\) onto \(\overline {a}\)) is given by \[Comp_{\overline {a}}\hspace {0.1cm}\overline {b} = \frac {\overline {a} \cdot \overline {b}}{\left |\overline {a}\right | }\]

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