(Untitled)

[burgundycolumns]

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (1, 2) and arrived in the Iron Hills at the point with coordinates (2, 6). 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

Comments

(hint pythagoras)

[bob]

if you want to try to figure out the angle from the vector you are given then you'll have all the angles (90, a, b) and one side (from (1,2) to (2,6)).

i think one or two applications of the pythagorean theorem a^2 + b^2 = c^2 should work. have you tried drawing it?

Re: (hint pythagoras)

[jcoveney]

This would also work but would probably have more ugly squared terms.

[jcoveney]

I found this method easiest.

First, since he turns at a right angle, we know that there is a vector that dotted with <3,2> equals 0. So equation one is

(x-2)3+(y-6)2=0

Equation two comes when you realize that the vector yields an equation which the point must be on

y=2/3*x+(2-2/3*1)

The point must be the intersection of these two points, or the point on the line he walks where the vector towards (2,6) yields a dot product of 0, or a right angle. Check by making sure that the point satisfies pythagoras.