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:Answer is x = -24/13,  y = 3/13Step-by-step explanation:The vector equation of line is;(x,y) = ([tex]x_{0},y_{0}[/tex]) + t (v)  ...............  1we know that;([tex]x_{0},y_{0}[/tex]) = (2,1)    andv = 5i + 1jput in equation 1[tex](x,y) = (2,1) + t (5i + 1j)[/tex]The parametric equation becomes;[tex](x,y) = 2i + 1j + 5ti + 1tj[/tex]by comparison we get;[tex]x = 5t + 2.................... 2\\y = t + 1................... 3[/tex]multiply equation 3 by -5 and add it to equation 2[tex]-5y = -5t - 5\\x = 5t + 2\\\\x - 5y = -3[/tex]we know that;[tex]m =(y_2 - y_1)/(x_2 -x_1)[/tex] 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:[tex]5x + y = c\\\\where \\x = 3 \\y = 6\\so,\\5(3) + 6 = c\\15 +6 =c\\21=c[/tex]So, the equation of other line is [tex]5x+y=21.......4\\x-5y = -3......5\\[/tex]multiply equation 5 by -5[tex]-5x+25y=-15\\5x+y=21\\\\26y=6\\y=6/26\\y=3/13[/tex]put this value of y in equation 5.[tex]x-5(3/13) =-3\\x-15/13=-3\\x=-3+\frac{15}{13}\\ x=\frac{-24}{13}[/tex]please rate 5 stars