Question

Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (2,1)(2,1) and arrived in the Iron Hills at the point with coordinates (3,6)(3,6). If he began walking in the direction of the vector v¯¯¯=5i+1jv¯=5i+1j and changed direction only once, when he turned at a right angle, at what point did he make the turn?

Answer 1

Answer is x = -24/13, y = 3/13

Explanation:

The vector equation of line is;

(x,y) = (
`x_(0),y_(0)`) + t (v) ............... 1

we know that;

(
`x_(0),y_(0)`) = (2,1)

and

v = 5i + 1j

put in equation 1

`(x,y) = (2,1) + t (5i + 1j)`

The parametric equation becomes;

`(x,y) = 2i + 1j + 5ti + 1tj`

by comparison we get;

`x = 5t + 2.................... 2\\y = t + 1................... 3`

multiply equation 3 by -5 and add it to equation 2

`-5y = -5t - 5\\x = 5t + 2\\\\x - 5y = -3`

we know that;

`m =(y_2 - y_1)/(x_2 -x_1)`

so, m = 5

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:

`5x + y = c\\\\where \\x = 3 \\y = 6\\so,\\5(3) + 6 = c\\15 +6 =c\\21=c`

So, the equation of other line is

`5x+y=21.......4\\x-5y = -3......5\\`

multiply equation 5 by -5

`-5x+25y=-15\\5x+y=21\\\\26y=6\\y=6/26\\y=3/13`

put this value of y in equation 5.

`x-5(3/13) =-3\\x-15/13=-3\\x=-3+(15)/(13)\\ x=(-24)/(13)`

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