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?

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:

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

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

Therefore the gradient of the new path walked

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

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

Therefore, the coordinates where he makes a turn is:

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