Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (−2,3) and arrived in the Iron Hills at the point with coordinates (−1,7). 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.

I'm stuck after doing the dot product. Thank you

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  1. no need for cross or dots I think
    turn at (a,b)
    what are the angles steered counterclockwise from x axis (I do ships but will try to stick to the math angles :)
    first leg
    tan Theta = 2/3 so theta = 37.4 degrees
    now what is the angle from start to finish
    delta y = 7 - 3 = 4
    delta x = -1 +2 = 2
    so tan angle = 4/2 = 2 so angle = 70.5 degrees
    we are there.
    in our triangle we have 70.5- 37.4 = 33 deg at start and 90 deg at the left turn
    so 90 - 33 = 57 degrees up at the finish
    we know the hypotenuse sqrt[(7-4)^2 +1^2] = sqrt 10
    cos 33 = length = that first leg / sqrt 10
    you now have the length of the first leg and the direction is 37.4 degrees above East so you can find the x y changes easy from (-2,3)

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