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 (-3, -1) and arrived in the Iron Hills at the point with coordinates (-1,3). If he began walking in the direction of the vector v = 5i + lj 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

Geometrically, the vector 5i + 1j corresponds to a slope of 1/5.
So, at the right angle where Gandalf changes direction, the slope becomes -5.

Let\'s write out the equations.
Point (-3, -1), slope 1/5 => y + 1 = (1/5)(x + 3) or y = (1/5)(x + 3) - 1 or y = (1/5)(x - 2)
Point (-1, 3), slope - 5 => y - 3 = (-5)(x + 1) or y = - 5x - 2.

Now, we find the intersecting point of these two curves
(1/5)(x - 2) = -5x - 2
==> x = - 4/13
Substituting this into either equation yields y = - 6/13.

So, the desired point is ( - 4/13, - 6/13).