For problems 1 – 4 below, let a⃗ = 〈3, −1,1〉, b+⃗ = 〈−2,3,9〉, and c⃗ = 〈5, −2, −1〉 and let k = −4 (a scalar). 1. Determine whether the following are vectors, scalars, or undefined. If the value is defined, use the values above to evaluate. If the value is undefined, clearly explain why. a. ka⃗ × 7b+⃗ ∙ c⃗9 b. kb+⃗ × kc⃗ c. 7a⃗ × b+⃗9 × 7b+⃗ × a⃗9 2. Verify that 7a⃗ + b+⃗9 ∙ 7a⃗ − b+⃗9 = ‖a‖! − ‖b‖! 3. Find the vector projection of c⃗ onto b+⃗. 4. Find the area of the parallelogram formed by a⃗ and c⃗. II. Proof-type problems 5. Do the following: a. Prove that if two vectors are orthogonal, then their dot product is zero. b. Show that a⃗ and b+⃗ (as defined in part I) are orthogonal. c. Explain why it must be the case that ka⃗ and kb+⃗ are orthogonal. 6. Complete one of the following: a. Use the definition of dot product to prove that u+⃗ ∙ u+⃗ = ‖u+⃗‖!. b. Prove that v⃗ × v⃗ = 0+⃗. c. (Extra Points for this one.) Suppose that ‖u+⃗‖ = 2 and ‖v‖ = 5 and ‖u+⃗ × v⃗‖ = 6. Find u+⃗ ∙ v⃗.