Question

Gandalf the Grey started in the Forest of Mirkwood at a point P with coordinates (2,−2) and arrived in the Iron Hills at the point Q with coordinates (3, 2). If he began walking in the direction of the vector v=5i+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?

Answer 1

(97/26; -43/26)

Explanation:

P=(2; -2) /*it is the initial position*/

Q=(3; 2) /*it's the final position*/

v=(5; 1) /*it's the initial velocity*/

Vn1=(-1; 5), Vn2=(1; -5) /*these are the orthogonal velocities, only one is the correct*/

X=? /*it is the point where I change the direction*/

r(t) /*it is the position vector*/

r(t)=P+v.t= (2; -2)+(5; 1)t = (2+5t; -2+t)

after I arrive to the position X, I'll change the direction.

so X=r(α)= (2+5α; -2+α), and after this:

r(t)=X+Vn.t => r(β)=(2+5α; -2+α)+Vn.β

I'll have 2 choices for Vn: Vn1 and Vn2, then:

1) Q=(3; 2)=(2+5α; -2+α)+(-1; 5)β= (2+5α-β; -2+α+5β)

{1=5α-β; 4=α+5β}

In this case: α=9/26 and β=19/26

2) Q=(3; 2)=(2+5α; -2+α)+(1; -5)β= (2+5α+β; -2+α-5β)

{1=5α+β; 4=α-5β}

In this case: α=9/26 and β=-19/26

α and β are instants of time, so, they can't be negative, then I have to discard the second choice.

So: X=(2+5α; -2+α), with α=9/26 => X=(97/26; -43/26)