Find all values of a if any so that the given vectors form a

Find all values of a (if any) so that the given vectors form an orthogonal set. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) u1 = 1 2 2 1 , u2 = 6 4 1 0 , u3 = 1 0 a 11

Solution

The vectors u₁ = [1 -2 2 -1]^T, u₂ = [6 4 1 0]^T, and u₃ = [-1 0 a 11]^T,

Now u₁.u₂ = 6 - 8 + 2 + 0 = 0. This implies that vectors u₁ and u₂ are orthogonal vectors.

Further, u₁.u₃ = -1 + 0 + 2a - 11 = 0 => a = 6, for which u₁ and u₃ are orthogonal vectors.

For a = 6, we have

u₂.u₃ = -6 + 0 + 6 + 0 = 0 => For a = 6, u₂ and u₃ are also orthogonal vectors.

Hence, for a = 6, the given set of vectors are orthognal vectors.