Vector problem?

Question

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (2, 1) and arrived in the Iron Hills at the point with coordinates (4, 5). 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.

I can do this by using equation of lines.. but I have no idea how to do this with vectors.

The answer is point (68/13, 41/13) using equation of lines, but I need to know how to do it using vectors

Answer 1

His journey is composed of two vectors, and .

He starts at (2,1) and ends at (4, 5) so the sum of the two vectors has got to be <4-2, 5-1>. That is:

x1 + x2 = 2
y1 + y2 = 4

But the two vectors are perpendicular. That means their dot product is 0. So

(x1)(x2) + (y1)(y2) = 0

Lastly we know the original direction, so:

(x1)/(y1) = 3/2

(In vector terms, = C<3,2> for some C, but there is no need to add an extra equation and an extra unknown)

These four equations are enough to compute all 4 values.

HTH.