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please give a complete solutionto question 3! only thank you!
Let u and v be unit vectors in R2. Our goal is to maximize u v. One way to do this is to interpret this as a constrained optimization problem: Note that |u| = 1 is equivalent to |u|2 = u u = 1. Let u = (x, y) and v = (z,w). Rewrite the above maximization problem in terms of x, y, z, w. Use Lagrange multipliers to show that u v is maximized provided u = v. Explain why the maximum value of u v must therefore be 1. Explain why for any unit vectors u and v we must have |u t| 1. Let u and u be any vectors in R2 (not necessarily unit). Apply your conclusion above to the vectors: u/|u| and v/|v| to show that (u v)2 |u|2|v|2. Show transcribed image text
please give a complete solutionto question 3! only thank you!
Let u and v be unit vectors in R2. Our goal is to maximize u v. One way to do this is to interpret this as a constrained optimization problem: Note that |u| = 1 is equivalent to |u|2 = u u = 1. Let u = (x, y) and v = (z,w). Rewrite the above maximization problem in terms of x, y, z, w. Use Lagrange multipliers to show that u v is maximized provided u = v. Explain why the maximum value of u v must therefore be 1. Explain why for any unit vectors u and v we must have |u t| 1. Let u and u be any vectors in R2 (not necessarily unit). Apply your conclusion above to the vectors: u/|u| and v/|v| to show that (u v)2 |u|2|v|2.
Show transcribed image text Reid WolffLv2
22 Apr 2019