Hi folks. This one has really stumped me. Other people have asked this question

Question

Hi folks. This one has really stumped me. Other people have asked this question on here before, but I still don't get it. Could someone please explain in simplest terms possible, how to solve this question?

Many thanks.

(1 pi) 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 (4, 3). If he began walking in the direction of the vector v = 4i + 1j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn.

Explanation / Answer

Geometrically the vector v = 4i + 1j has a slope of 1/4

So, at the right angle where Gandalf changes direction, the slope becomes -4.

Let's write out the equations.

He starts at (2 , -1) with a slope of 1/4

So, y - y1 = m(x - x1)
y - (-1) = (1/4)(x - 2)
y + 1 = (1/4)x - 1/2
y = (1/4)x - 1/2 - 1
y = (1/4)x - 3/2 ----> first equation

Now, he ends up at the point (4,3) with a slope of -4

So, y - y1 = m(x - x1)
y - 3 = -4(x - 4)
y - 3 = -4x + 16
y = -4x + 19 ---> second equation

Solving first and second equations will give us the intersection, i,.e the point where he turned

(1/4)x - 3/2 = -4x + 19

Multiply all over by 4 :

x - 6 = -16x + 76

17x = 82

x = 82/17

y =- 4x + 19
y = -4(82/17) + 19
y = -5/17

So, the point is :

(82/17 , -5/17) ----> ANSWER

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