Vector Coordinates

Question

Grey started in the Forest of Mirkwood at a point P with coordinates (3,-3) and arrived in the Hills at point Q (4,1). If he began walking in the direction of the vector v = 3i +2j and changes the direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn?

If you could explain how you get your answer, that would be helpful!

Answer 1

There are many ways to do this. A purely algebraic way is to realize that the line that is perpendicular to 3i + 2j is in the direction -2i + 3j (just taking the reciprocal slope). You can equate the two lines that go through the two points to find the intersection (which is where the right angle turn must occur):

line through P and slope of 2/3 : y+3 = (2/3)(x-3) or y = (2/3)x - 5

line through Q and slope of -3/2: y -1 =(-3/2)(x-4) or y = -(3/2)x +7

Equate the two equations and solve for x. Plug into either equation for y.

(I get 72/13 for x)

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