Main Concept

Recall that the vector projection of a vector $\stackrel{\rightharpoonup }{u }$ onto another vector $\stackrel{\rightharpoonup }{v }$ is given by ${\mathrm{proj}}_{\stackrel{\rightharpoonup }{v }}\left(\stackrel{\rightharpoonup }{u }\right) \= \frac{\stackrel{\rightharpoonup }{u }·\stackrel{\rightharpoonup }{v }}{{∥\stackrel{\rightharpoonup }{v }∥}^2}\stackrel{\rightharpoonup }{v }$.

The projection of $\stackrel{\mathbf{\rightharpoonup }}{\mathit{u}\mathbf{ }}$ onto a plane can be calculated by subtracting the component of  $\stackrel{\rightharpoonup }{u }$ that is orthogonal to the plane from  $\stackrel{\rightharpoonup }{u }$.  If you think of the plane as being horizontal, this means computing  $\stackrel{\rightharpoonup }{u }$ minus the vertical component of $\stackrel{\rightharpoonup }{u }$, leaving the horizontal component. This "vertical" component is calculated as the projection of  $\stackrel{\rightharpoonup }{u }$ onto the plane normal vector $\stackrel{\rightharpoonup }{n }$.

${\mathrm{proj}}_{\mathrm{Plane}}\left(\stackrel{\rightharpoonup }{u }\right) \= \stackrel{\rightharpoonup }{u } - {\mathrm{proj}}_{\stackrel{\rightharpoonup }{n }}\left(\stackrel{\rightharpoonup }{u }\right) \= \stackrel{\rightharpoonup }{u } - \frac{\stackrel{\rightharpoonup }{u }·\stackrel{\rightharpoonup }{n }}{{∥\stackrel{\rightharpoonup }{n }∥}^2}\stackrel{\rightharpoonup }{n}$

More MathApps

MathApps/NaturalSciences