Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (2,1) and arrived in the Iron Hills at the point with coordinates (3,5). If he began walking in the direction of the vector v=4i+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.

The line that makes a right angle with another is called perpendicular, and it has its slopes equal to the opposite of the inverse of the slope of the other line, so:

slope = - 2

The point (x,y) also pertences to it, such as the point (3,5), so:

-2 = (y - 5)/(x - 3)

y - 5 = -2x +6

y = -2x + 11

So, the turn point is the point where these two lines meet:

x/2 = -2x + 11

x = -4x + 22

5x = 22

x = 22/5

y = (22/5)/2

y = 22/10 = 11/5

So, the point that he makes the turn is (22/5, 11/5).

AFOKE88 AFOKE88 · Answer 2 · 2022-04-08T10:11:58+02:00

The coordinates of the point where he makes the turn at a right angle are gotten as; (22/5, 11/5)

What are the coordinates of the point?

From the given coordinates, we can deduce the parametric equations as;

x = 4t + 2

y = 2t + 1

The standard Cartesian form can be found by multiplying y equation by -2 and adding it to the x equation to get;

x - 2y = 0 ----(1)

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 and so; 2x + y = c

We can find the value of c by forcing the line to contain the point (3,5)

2(3) + 5 = c

c = 11

Equation of other line is;

2x + y = 11 ------(eq 2)

Putting 2y for x in eq 2 gives;

2(2y) + y = 11

5y = 11

y = 11/5

x = 2(11/5)

x = 22/5

Read more about prametric equations at; https://brainly.com/question/10165611