Need help with this Calculus 3 Question.

Question

Gandalf the Grey started in the Forest of Mirkwood at a point P with coordinates (−3,1) and arrived in the Iron Hills at the point Q with coordinates (-2, 4). 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

Hint:

So $(-3,1)+t(5,1)+s(-1,5)=(-2,4)$.

Hence there is a linear system of two equations in two unknowns.

Solve.

Then your point is $(-3,1)+t(5,1)$.

Answer 2

$Q-P = (-2,4) - (-3,1) = (1,3)$

Plotting these points on the complex plane, multiplication by $i$, makes for a 90 degree turn to the left.

$(a+bi)(1+5i) = (1+3i)\\ a+bi = \frac{1+3i}{1+5i}\\ a+bi = \frac{(1+3i)(1-5i)}{1+5^2}\\ a+bi = \frac{16 - 2i}{26}$

The turning point is at:

$P + a(1,5)$

Answer 3

Let the point of turn be $(a,b)$ which is along the starting path given by, i.e.

$$b-1=\frac15(a+3)\tag 1$$

Given that the turn is 90-degrees, we have the slope equation

$$\frac{a+3}{b-1}=-\frac{b-4}{a+2}\tag 2$$

Solve the system (1)-(2) for the solutions $(-3,1)$ and $(-\frac{19}{13},\frac{17}{13})$, of which only the latter is valid turning point.