# Gandalf's adventure (simple vector algebra)

So, I found the correct answer to this homework question, but I was hoping there was an easier way to find the same answer.

Here's the 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 $(-1, 6)$. If he began walking in the direction of the vector $\bf v = 5 \mathbf{I} + 1 \mathbf{J}$ and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn.

The answer is $((-1/13), (18/13))$.

Now, I know that the dot product of two perpendicular vectors is $0$, and the sum of the two intermediate vectors must equal $\langle 1, 5\rangle$. Also, the tutor solved the problem by using a vector-line formula which had a point, then a vector multiplied by a scalar. I'm looking for the easiest and most intuitive way to solved this problem.

Any help is greatly appreciated! I'll respond as quickly as I can.

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His first leg is $a(5,1)$ and second leg is $b(1,-5)$ (because it is perpendicular to the first) with total displacement $(1,5)$. So

$5a+b=1$

$a-5b=5$

Then the turning point is $(-2,1)+a(5,1),$ which should equal (here is a check) $(-1,6)-b(1,-5)$

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Thanks! in class but will check in two hours! –  user13327 Sep 12 '11 at 23:40
Awesome that was way easy! Thanks! –  user13327 Sep 13 '11 at 13:43

The best way I can see it is to write an equation for the line of Galdalf's path $p$. Since he walks in the $\langle 5,1\rangle$ direction, such a path will have slope $1/5$ in $\mathbb{R}^2$. Since he starts at $(-2,1)$, $p$ can be described by $$p=\frac{1}{5}x+\frac{7}{5}.$$ To find the turning point, you want to project $(-1,6)$ onto the path, since the turning point is supposed to be at a right angle. I always have a hard time recalling the projection formulas, so an easy way to remember is that this projection must have slope $-5$ (in order to be perpendicular), and since it passes through $(-1,6)$ has equation $$p'=-5x+1.$$ The turning point is now just the intersection. Calculating, $$\frac{1}{5}x+\frac{7}{5}=-5x+1\implies x=\frac{-1}{13},\ y=\frac{18}{13}.$$

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:D thanks mate! –  user13327 Sep 12 '11 at 23:53