Question

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (2,−2) and arrived in the Iron Hills at the point with coordinates (3,2). If he began walking in the direction of the vector v=4i+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.

Answer 1

This problem is basically asking you to find a perpendicular vector to Gandalf's initial path. The direction if this vector will be set no matter when he turns so we can start by calculating that.

Vectors are perpendicular if their dot product is 0, so we need <x,y> such that 5x + 2j = 0. The obvious choice will be <-2,5>, as that clearly satisfies our conditions (this is of course identical to <2, -5>).

So now we have reduced this problem to finding where the line that goes through (1,6) with a slope of -2.5 intersects our original line, which we can easily see is y = 0.4x + 3.

The equation for the new line can be seen to be y = -2.5x + 8.5.

Setting these equal,

0.4x + 3 = -2.5x + 8.5

2.9x = 5.5

x = 5.5/2.9 = 1.896, plugging back in yields y = 3.7586

This approach sort of reduces the problem to just a planar geometry exercise, which may or may not be desirable, but it surely yields the correct result.