Answer Happy

Vi 0-- V2 , V3 0 1 2 1 2 Consider the weighted Euclidean inner product on R4 given by (u, w) = 2u1W1 + U2W2 + U3w3 +3U4W4 where u; and w; for i = 1, 4 are the coordinates of the vectors u and w respectively. (a) Determine the cosine of the angle between any two of the above vectors, where the angle is considered with respect to the standard Euclidean inner product. Which vectors are orthogonal? (b) Determine the cosine of the angle between any two of the above vectors, where the angle is considered with respect to the weighted Euclidean inner product defined above. Which vectors are orthogonal? (c) Find two vectors in R4 that are orthogonal with respect to both the standard Eu- clidean inner product and the weighted Euclidean inner product defined above. (d) Check that the Cauchy-Schwarz inequality holds for vi and V2 with respect to the standard Euclidean inner product and with respect to the weighted Euclidean inner product defined above. (e) Find a basis for the orthogonal complement of the subspace of R4 spanned by V1, V2 and V3 with respect to the standard Euclidean inner product.