Question

Gandalf the Grey started in the Forest of Mirkwood at a point P with coordinates (3, 0) and arrived in the Iron Hills at the point Q with coordinates (5, 5). If he began walking in the direction of the vector v - 3i + 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?

Answer 1

Answer:

Turning point has coordinates $\left(\dfrac{27}{13},\dfrac{8}{13}\right)$

Step-by-step explanation:

Gandalf the Grey started in the Forest of Mirkwood at a point P(3, 0) and began walking in the direction of the vector $\vec{v}=-3i+2j.$ The coordinates of the vector v are (-3,2). Then he changed the direction at a right angle, so he was walking in the direction of the vector $\vec{u}=2i+3j$ (vectors u and v are perpendicular).

Let B(x,y) be the turning point. Find vectors PB and BQ:

$\overrightarrow{PB}=(x-3,y-0)\\ \\\overrightarrow {BQ}=(5-x,5-y)$

Note that vectors v and PB and vectors u and BQ are collinear, so

$\dfrac{x-3}{-3}=\dfrac{y}{2}\\ \\\dfrac{5-x}{2}=\dfrac{5-y}{3}$

Hence

$2(x-3)=-3y\Rightarrow 2x-6=-3y\\ \\3(5-x)=2(5-y)\Rightarrow 15-3x=10-2y$

Now solve the system of two equations:

$\left\{\begin{array}{l}2x+3y=6\\ -3x+2y=-5\end{array}\right.$

Multiply the first equation by 3, the second equation by 2 and add them:

$3(2x+3y)+2(-3x+2y)=3\cdot 6+2\cdot (-5)\\ \\6x+9y-6x+4y=18-10\\ \\13y=8\\ \\y=\dfrac{8}{13}$

Substitute it into the first equation:

$2x+3\cdot \dfrac{8}{13}=6\\ \\2x=6-\dfrac{24}{13}=\dfrac{54}{13}\\ \\x=\dfrac{27}{13}$

Turning point has coordinates $\left(\dfrac{27}{13},\dfrac{8}{13}\right)$