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 (3,5). 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

Answer:

(22/5, 11/5)

Step-by-step explanation:

Gandalf starts walking in the vector v = 4i + 2j, the slope of it must be 2/4 = 1/2.

So, calling the turn point (x,y) it must be in that vector, and the start point (2,1) must also be in this line, so:

slope = Δy/Δx

1/2 = (y - 1)/(x - 2)

2y - 2 = x - 2

2y = x

y = x/2

The line that makes a right angle with another is called perpendicular, and it has its slopes equal to the opposite of the inverse of the slope of the other line, so:

slope = - 2

The point (x,y) also pertences to it, such as the point (3,5), so:

-2 = (y - 5)/(x - 3)

y - 5 = -2x +6

y = -2x + 11

So, the turn point is the point where these two lines meet:

x/2 = -2x + 11

x = -4x + 22

5x = 22

x = 22/5

y = (22/5)/2

y = 22/10 = 11/5

So, the point that he makes the turn is (22/5, 11/5).