Vector Projection Length

Original Post

Apr 5, 2009 at 5:10 PM

Im a little confused with regards to the follwoing equations for the length of the projection. The one below makes sense,

length = |p| cos(0) p.q = |p| |q| cos(0) p.q / |q| = |p| cos(0) length = p.q / |q| But i cant seem to fiqure out the following one. Why the divide by |w| when using the trig approach? i dont see why that is needed.

Bob Janova

Apr 5, 2009 at 6:03 PM

I guess it's showing you how to project into units of w? It isn't clear though. The length (in world coordinates) of that line segment is |v| cos theta.

Doggan

Apr 6, 2009 at 2:21 PM

The 2nd figure is showing you how to calculate a dot product via trig.

cos(theta) = |w| / |v|
|v| cos(theta) = |w|

...but this is assuming that the projection of v onto w is w (i.e. v.w == |w|), which is not always valid. In the figure, |w| is actually less than the projection. So you need to account for this:

|v| cos(theta) = |w| * v.w
|v| cos(theta) / |w| = v.w

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