am having trouble setting up how to do the problem. To get from point
(3,2) to (4,7) starting with a vector 5i+2j and turning once at 90 degrees
to get to that specific endpoint. I know that in order to be orthogonal
the dot products of the 2 vectors needs to be zero....but I don't know what
2 vectors to use and then once I have the two vectors to use....do I just
subtract or add one of those vectors to the start or end point...? Please
help!
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OK the direction of the Iron Hills from the forest of Mirkwood is <1,5>=<4,7>-<3,2>. Good old Gandalf doesn't go that way though (because that would be too simple). Instead he goes in the direction of the vector <5,2> for awhile (which is *not* the same direction of <1,5>), and after awhile takes a single right turn and heads in that direction and ends up at <1,5> (I emphasize the words "right turn" because that is the crucial bit of information that tells us what we need to do)
So the picture looks like this:
If you put it in this way, it should gradually become clear that this is a problem about orthogonal projection. The right pointing red arrow is the orthogonal projection of <1,5> on <5,2>, i.e.
=(<1,5>.<5,2>)/(<5,2>.<5,2>)<5,2>=15/29<5,2>=<75/29,30/29>
This is the point where he makes his turn.
The upward pointing red arrow is the difference or residual <1,5>-<75/29,30/29>=<-46/29,115/29>.
