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265Lesson02Problems

# 265Lesson02Problems - h 2 1 i and h-3 1 i in the xy-plane...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 2 PROBLEMS Remember: the dot and the cross used to denote the dot and cross products are not optional. Problems are worth 20 points each. (1) Compute h 3 , 1 , 0 i · (2 h- 1 , 1 , 2 i ). (2) Calculate the angle between -→ v = h 2 , 2 , 4 i and -→ w = h 2 , - 1 , 1 i . The result is a familiar angle, so the answer is to be given exactly. (3) Compute h 3 , 1 , 0 i × h- 1 , 1 , 2 i . (4) Find two unit vectors perpendicular to both -→ v = h 3 , 1 , - 1 i and -→ w = h 0 , 1 , 2 i . See Theorem 1, part (i), page 685. Don’t forget you are asked for unit vectors. Also don’t forget that -→ ı , -→ , and -→ k are vectors, and so they must have the arrow decoration . Once you have found one of the unit vectors, stop and think about why it is really easy to find the second one. (5) Sketch the parallelogram spanned by
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Unformatted text preview: h 2 , 1 , i and h-3 , 1 , i in the xy-plane, and compute its area. See Example 6, page 688. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Verify the Distributive Law (Theorem 1, page 676). Example 1, page 676, veriﬁes the law for three particular vectors. But now the idea is to verify it for all vectors. So, let-→ u = h a 1 ,b 1 ,c 1 i ,-→ v = h a 2 ,b 2 ,c 2 i , and-→ w = h a 3 ,b 3 ,c 3 i . Then, just as in Example 1, page 676, ﬁrst compute-→ u · (-→ v +-→ w ), then compute-→ u ·-→ v +-→ u ·-→ w , and then compare the results to make sure they are the same. 1...
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