10.4#8

Hello, Dr. Taylor.

I solved this problem by projecting vector QP onto vector QR, finding the angle measure between vectors QP and QR with the arccos function, and then calculating sqrt(14)sin(theta) to find the
distance d.  However, I am also aware that using cross products rather than projections is the focus of this lesson - how can this problem be solved using cross products? Using the formula ||U x V|| = ||U|| ||V|| sin(theta), I would end up with 14sin(theta) instead of sqrt(14), since both QP and QR
have magnitudes of sqrt(14).

Thanks for your assistance.











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OK, some short answers are 1) you started on one possible right track using projections, but then you got off the trail and started bushwhacking through the weeds, and 2) that nice formula
||U x V|| = ||U|| ||V|| sin(theta)
you use comes in three pieces, it's up to you to figure out how to take the pieces you want.

But when doing these problems I think it's important to take an organized approach, and the beginning should be TO DRAW A PICTURE OF THE PROBLEM from a perspective that captures the essential features.  This three-D plot captures EVERYTHING:

but maybe it's too much information?  This simpler diagram says what we need to know:

See that right triangle? You know the size of the hypotenuse, and the piece || to QR is the projection so you know it's size.  The other piece is the answer you need.  You can get that in several ways. Hint: cos^2(x)+sin^2(x)=1.