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 (5,2). If he began walking in the direction of the vector v=3i+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

for vector ai+bj,slope is b/a

Therefore for vector v=3i+ 1j,slope=1/3

And Gandalf the Grey walking in the direction of vector,therefore slope of Gandalf and Grey journey is 2/3 and it passes through(3,-1)

Therefore we get

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

y=x/3   - 2

And the slope of the perpendicualr line is m =1/3 and it passes through (5,2)

y-2=3(x-5)

y=3x-13

At intersection or turning point

x/3 - 2=3x-13

x=33/8

Therefore y= x/3 - 2= 33/8/3 - 2 = -5/8

Therefore the required point is (33/8,-5/8)