If he began walking in the direction of the vector v=4i+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.

Question

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (0, -3) and arrived in the Iron Hills at the point with coordinates (2, 2) . If he began walking in the direction of the vector v=4i+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.

Answer 1

The slope of the vector 4i + 2j = rise/run = 2/4 = 1/2 making the equation of the line y = 1/2x - 3

A right angle turn would result in the line of direction having a slope of -2 and the resulting equation of the line (in point slope form) being y - 2 = -2(x - 2) or y = -2x + 6

The intersection point of y = 1/2x - 3 and y = -2x + 6 is found by equating the right sides of the equations:

1/2x - 3 = -2x + 6

1/2x = -4/2x + 9

5/2x = 9

x = 9(2/5) = 18/5

y = -2(18/5) + 6

y = -36/5 + 30/5

y = -6/5

So the intersection point is (18/5, -6/5)