Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (-1, 3) and arrived in the Iron Hills at the point with coordinates (1, 6). If he began walking in the direction of the vector v = 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?
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The equation of the line passing through (-1, 3) in the direction of <4, 1> is
r(t) = (-1, 3) + t(4, 1) = (4t - 1, t + 3).
A vector perpendicular to <4, 1> is <-1, 4> (or any scalar multiple thereof).
So, the equation of the line passing through (1, 6) in the direction of <-1, 4> is
R(s) = (1, 6) + s(-1, 4) = (-s + 1, 4s + 6).
Now, all we need to do is see where these lines intersect.
Equating x and y entries yields
4t - 1 = -s + 1 ==> s + 4t = 2
t + 3 = 4s + 6 ==> 4s - t = -3.
Solving for s and t yields s = -10/17 and t = 11/17.
Substituting s or t into either equation yields the point of intersection
(x, y) = (27/17, 62/17).
I hope this helps!
r(t) = (-1, 3) + t(4, 1) = (4t - 1, t + 3).
A vector perpendicular to <4, 1> is <-1, 4> (or any scalar multiple thereof).
So, the equation of the line passing through (1, 6) in the direction of <-1, 4> is
R(s) = (1, 6) + s(-1, 4) = (-s + 1, 4s + 6).
Now, all we need to do is see where these lines intersect.
Equating x and y entries yields
4t - 1 = -s + 1 ==> s + 4t = 2
t + 3 = 4s + 6 ==> 4s - t = -3.
Solving for s and t yields s = -10/17 and t = 11/17.
Substituting s or t into either equation yields the point of intersection
(x, y) = (27/17, 62/17).
I hope this helps!