Help with this weird calc question?

Question

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(1, 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.

Guy · Accepted Answer

Okay...Gandalf starts at (-1, 2), and walks toward a point that is four units to the right, and 2 units up; he's walking toward the point (3, 4).

The equation of the line that goes through the points (3, 4) and (-1, 2) is y = 0.5x + 2.5

We need to find the point on the line that has the shortest distance to (1, 5).  This will naturally be at a right angle, so we need not worry about that. Using the distance formula, we have:

d^2 = (x - 1)^2 + (y - 5)^2

d^2 = (x - 1)^2 + (0.5x + 2.5 - 5)^2

d^2 = (x - 1)^2 + (0.5x - 2.5)^2

d^2 = (x^2 - 2x + 1) + (.25x^2 - 2.5x + 6.25)

d^2 = 1.25x^2 - 4.5x + 7.25

d^2 (and therefore, d), will be a minimum at x = -b/(2a) = 4.5/(2*1.25) = 1.8

The y-coordinate will be y = 0.5(1.8) + 2.5 = 3.4

Answer: (1.8, 3.4), or with fractions, it's (9/5, 17/5).

Pete B · Answer

I am not sure if I am solving this the right way but I just graphed the coordinates and drew the <4,2> vector. From (-1,2) at a vector <4,2> the line goes through the point (0,5). If a right turn was made at a right angle there, then it would reach (1,5). I am sorry if this is not correct or not the correct way of doing it, but I am just trying to help.

bskelkar · Answer

What is calc? what it has to here? tHIS IS A COORDINATE GEOMETRY PROBLEM. I think you are misguided.
 Use a graph paper, its easy by that method.

Help with this weird calc question?

3 Answers