Figuring out the Orthogonal Coordinates from a a given point and vector

Question

I am given this Question of

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (-2, -3) and eventually arrived in the Iron Hills at the point with coordinates (0, 0). 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.

I have absolutely no idea where I would start. I've tried using the distance formula, tried finding the slope and plugging in the points and using the inverse of the slope to find out the x and y but nothing works

You mention the “inverse of the slope.” What exactly do you mean by that? — amd, Sep 08 '18 at 07:47
I wonder if saying "You shall not pass!" would be in poor taste here? :) — Deepak, Sep 08 '18 at 08:36

score 1 · Answer 1 · answered Sep 08 '18 at 07:59

Gandalf wants to travel from $(-2, -3)$ to $(0,0)$.

He is traveling along the direction $(4,2)$ and $(-2,4)$.

Hence we just have to solve for $$(-2,-3) + x(4,2) + y(-2,4) =(0,0)$$

The turning point is $(-2,-3)+x(4,2)$.

Solving for $x,y$ is just solving two linear equations in $2$ unknowns and the solution is unique.

Figuring out the Orthogonal Coordinates from a a given point and vector

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