Gandalf the Grey started in the Forest of Mirkwood at a poin

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, 3). If he began walking in the direction of the vector v = 4i + 1j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn.

Solution

First, he travels in direction of vector v = (1,4)
Then he turns at a right angle and travels in direction of vector u = (-4,1)

m(1,4) + n(-4,1) = (3,3) - (2, 0)
(m-4n, 4m+n) = (1,3)

m - 4n = 1 . . . . . m - 4n = 1
4m + n = 3 . . . . . 16m +4 n = 3

17m = 4

m = 4/17

Point where Gandalf turns = (2, 0) + 4/17 (1,4) = (38/17, 16/17)