Let v = (v1, v2) be a vector in r2. show that (v2, −v1) is orthogonal to v, and use this fact to find two unit vectors orthogonal to the given vector. v = (9, 40)
Accepted Solution
A:
Because v1 x v2 + v2 x (-v1) = v1 x v2 - v2 x v1 = v1 x v2 - v1 x v2 = 0, vector (v1,v2) is orthogonal to (v2,-v1); Let (a,b) be an unit vector orthogonal to (9,40); So, a^2 + b^2 = 1 and a x 9 + b x 40 = 0; Then, a = - 40 x b / 9; Finally, 160 x b^2 / 81 + b^2 = 1; 160 x b^2 + 81 x b^2 = 81; 241 x b^2 = 81; b^2 = 81 / 241; b = + or - 9/[tex] \sqrt{241} [/tex] a = + or - 40/[tex] \sqrt{241} [/tex] We have (+40/[tex] \sqrt{241} [/tex];9/[tex] \sqrt{241} [/tex]) and (-40/[tex] \sqrt{241} [/tex];-9/[tex] \sqrt{241} [/tex]);