Let R4 have the Euclidean inner product Find a vector in R4

Let R^4 have the Euclidean inner product. Find a vector in R^4 that is orthogonal to u_1 u_4= and makes equal angles with u_2 u_s. Find a vector x = (x_1. x_2. x_3 x_4) of length 1 that is ortho above and such that the cosine of the angle between x an cosine of the angle between x and u_3.

Solution

(a). Let x = (x₁ , x₂ , x₃ , x₄ ) be the required vector. Since x is orthogonal to u₁ and u₄, we have x.u₁ = 0 and x.u₄ = 0 or, x₁ = 0 and x₄ = 0 so that x=(0, x₂ , x₃ , 0). We also know that for any two real vectors v and w, v.w = |v|*|w| cos , where is the angle between v and w. Let the angles between x and u₂ and also between x and u₃ be . Then x.u₂ = IxI*Iu₂I cos and x.u₃ = IxI*Iu₃I cos or, x₂ = IxIcos and x₃ = IxIcos as Iu₂I = Iu₃I = 1. Therefore x₂ = x₃ = t (say). Then x = ( 0, t, t,0) = t ( 0,1,1,0). Thus, the required vector is ( 0,1,1,0).

(b).Let x = (x₁ , x₂ , x₃ , x₄ ) be the required vector.Then , as above, since x is orthogonal to both u₁ and u₄ , we have x₁ = 0 and x₄ = 0 so that x =(0, x₂ , x₃ , 0).Let and Ø be the angles between x and u₂, and x and u₃ respectively. Then as above, x.u₂ = IxI*Iu₂I cos and x.u₃ = IxI*Iu₃IcosØ or, x₂ = cos and x₃ = cosØ as IxI = Iu₂I = Iu₃I = 1. Now, since cos = 2 cosØ, we have x₂ = 2x₃. Let x₃ = t. Then x = ( 0, 2t,t,0) = t ( 0,2,1,0). Therefore, the required vector is ( 0,2,1,0).