Part 6: Projections

1. Use the applet below to investigate the projection of a vector v (green) onto another vector w (yellow ) in space, which we will relate shortly to the dot product <v,w>. You can adjust the components of the yellow and green vectors and then ask the applet to compute the projection of the green vector onto the yellow. The slider at the bottom allows you to rotate the view of the vectors. To help keep you oriented, the projections of the green and yellow vectors onto the xy-plane are also shown. The projection of a vector v onto w is a vector in the same or opposite direction as w, so there are two things to be determined: "same or opposite" and the length of the projection. Think about how both of these might be related to the dot product.

We will use the idea of projecting one vector onto another to resolve a vector v into its components parallel and perpendicular to a vector w. The idea is illustrated in the following figure, which is drawn in the plane determined by v and w.

In this figure, v is shown in green and w in blue (partly obscured by u and p). The vector p is the projection of v on w, and this is also the component of v parallel to w. The component of v perpendicular to w is q, and clearly v = p + q. Thus, if we can find p from v and w, then we can calculate q as q = v - p.

2. For purposes of indicating the direction of w, we first calculate a unit vector u (i.e., u has length 1) in the same direction as u. How is u calculated from w? Explain why p = |p| u. How is |p| related to the length of v and the cosine of the angle between them?
3. Observe that cos can be calculated by taking the dot product of v with either of the vectors w or u. (We can't use p for this purpose because we don't know what it is yet -- that 's what we're trying to find.) Now put all the pieces together to arrive at a formula for p by finding u, cos , |p|, and finally p itself.
4. Carry out your formula to find the projection of v = (2,1,3) onto w = (2,4,2). Then find both the parallel and perpendicular components of v relative to w. If your calculation is correct, the two components should be perpendicular -- check that.
5. Use your formula for p to show that p and v - p are always perpendicular.

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Copyright CCP and the author(s), 2001-2002