Homework problems 1. Verify the following identities for vectors v, w in R": (a) (av) (bw) = ab(v· w) (b) (av+bu) = a(u⋅ v) +b(u⋅ w) (c) |vw|² = |v|2 + |w² + 2 v⋅ w (d) v+w² – vw|² = 4v · w - (e) |√+w² + |v – w² = 2|v|² + 2 ||² 2. Verify the following "if and only if" statements for vectors u, w in R": (a) = 0 if and only if |+| = |√ – w| (b) = 0 if and only if |ʊ + w|² = |v|² + |w|² (c) vw=0 if and only if + cш| ≥ || for all values of the scalar CER (d) (v+w) (v– w) = 0 if and only if |σ| = ||| . 3. Assuming both and w are nonzero vectors, define their angle to be 0 = arccos ช.พี (a) Verify that the quantity inside the arccosine function lies in the domain [-1, +1] of that function. (b) Verify the following identity: | |√ – w|² = |v|² + ||² – 2 || || cos 0 (c) Why is this identity the same thing as the old "law of cosines" that you learned in trigonometry?