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 (2,3) and arrived in the Iron Hills at the point with coordinates (3,2). If he began walking in the direction of the vector v=3i+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.

Solution

His first leg is a(3,2) and second leg is b(2,-3) (because it is perpendicular to the first)

with total displacement (1, 5).

So 3a +2b = 1

2a -3b =5

solve the two equation.

a =1 ; b=-1

Turning point : (3, 3a, 2 +2a) = (3 +1(3) , 2 +1(2) ) = (6, 4)