Finding the coordinates a vector turns 90 degrees to reach the final point?

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates #(1, -1)# and arrived in the Iron Hills at the point with coordinates #(2, 3)#. If he began walking in the direction of the vector #vecv=4hati+1hatj# and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn.

1 Answer
Douglas K.
Jan 25, 2018

The point is #(49/17,-9/17)#

Explanation:

The vector equation of a line is:

#(x,y) = (x_0,y_0) + t(vecv)#

In our case, #(x_0,y_0) = (1,-1)# and #vecv= 4hati+1hatj#

#(x,y) = (1,-1) + t(4hati+1hatj)#

The parametric equations are:

#x = 4t+ 1#
#y = t-1#

The standard Cartesian from can be found by multiplying y equation by -4 and adding it to the x equation:

#x - 4y= 5#

We know that we can find the family of lines that are perpendicular to the above line by swapping the coefficients and change the sign of one of them:

#4x+y=c#

We can find the value of c by forcing the line to contain the point #(2,3)#:

#4(2)+(3)=c#

#c = 11#

The equation of the other line is:

#4x+y=11#

The turning point is at the intersection of the lines:

#x - 4y= 5#
#4x+y=11#

#17x+ 0y = 49#

#x = 49/17#

#y = (5-49/17)/-4#

#y = -9/17#

The point is #(49/17,-9/17)#

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