5. Let  be a real inner product space and let and be nonze…

5. Let  be a real inner product space and let and be nonzero vectors in . Let be the vector subspace of consisting of all scalar multiples of the vector , and define the orthogonal projection from into by  a) Determine the value of the real scalar  that minimises the quantity . (Hint: expand as a quadratic in .)   b) Using your minimizing value of , show that .   c) Deduce the Cauchy–Schwarz inequality              and state precisely when equality holds. (Hint: try rewriting Cauchy–Schwarz as .)      How can you use b) to finish?