Hello Mr. Taylor:
I have been having problems with this:
"Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (2, 0)
and arrived in the Iron Hills at the point with coordinates (3, 4). If he began walking in the direction of the vector v=5i+2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. "
I think that it is a vector addition problem (the parallelogram rule?). However, I have no idea how to begin solving it. I know that I have one out of the two vectors, but I do not know how to start
Thank you!
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(Apologies for the late reply, I already answered this once, but it looks like I didn't save correctly.) Sure, this is a vector addition problem, but it's also a problem about the dot product (this is section 10.3 after all). So let's rephrase the problem: you start at the point (2,0), then you need to move in the direction of the vector 5i+2j some unknown amount, and then there is another vector xi+yj which you don't know except that it is orthogonal to 5i+2j (the word "orthogonal" is the clue that tells you need to use the dot product somehow), and then go along that perpendicular direction for another unknown amount until you end at the point (3,4). So here's:
RECIPE
1)Figure the total displacement vector by subtracting the initial point from the final point.
2) Use the dot product to figure out the projection of the displacement vector on the vector 5i+2j. This will tell you how far you need to go along the direction 5i+2j.
3) subtract the projection from the displacement; this is perpendicular to 5i+2j (how would you check this?).
4) the initial point plus the projection of the displacement is the point you need (why?)
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