Question

Gandalf the Grey started in the Forest of Mirkwood at a point PP with coordinates (2,−1)(2,−1) and arrived in the Iron Hills at the point QQ with coordinates (3, 3). If he began walking in the direction of the vector v=4i+1jv=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?

Answer 1

Answer:

$(\frac{66}{17},-\frac{9}{17})$

Step-by-step explanation:

Gandalf's Starting Point is given as: P(2,-1)

If he walks in the direction of the vector v=4i+1j

x=4, y=1

The slope of the line which he walked therefore is: $m_1=\dfrac{1}{4}$

First, we determine the equation of the line at P with coordinates (2,-1)

$y-y_1=m(x-x_1)\\y-(-1)=\frac{1}{4}(x-2)\\ y+1=\frac{1}{4}x-\frac{1}{2}\\y=\frac{1}{4}x-\frac{1}{2}-1\\y=\frac{1}{4}x-\frac{3}{2}$

If he changes direction at a right angle, the new path walked is perpendicular to the old path.

DEFINITION: Two lines are perpendicular if the product of their gradients$m_1m_2=-1$

Therefore the gradient $m_2$ of the new path walked $=-4$

At point Q with coordinates (3,3), the equation of the line is:

$y-y_1=m(x-x_1)\\y-3=-4(x-3)\\y=-4x+12+3\\y=-4x+15$

The coordinate where Gandalf the Grey makes a turn is the intersection of the two lines.

$If \: y=\frac{1}{4}x-\frac{3}{2} \:and\: y=-4x+15\\Then:\\\frac{1}{4}x-\frac{3}{2} =-4x+15\\\frac{1}{4}x+4x=15+\frac{3}{2}\\4.25x=16.5\\x=\frac{66}{17}\\y=-4x+15=-4(\frac{66}{17})+15=-\frac{9}{17}$

Therefore, the coordinates where he makes a turn is:

$(\frac{66}{17},-\frac{9}{17})$