27. a. Cauchy-Schwartz inequality Since u · v = |u||v| cos 0, show that the inequality Ju•v| < |u||v| holds for any vectors u and v. b. Under what circumstances, if any, does |u• v| equal |u||v|? Give reasons for your answer. 28. Dot multiplication is positive definite Show that dot multipli- cation of vectors is positive definite; that is, show that u ·u 2 0 for every vector u and that u · u = 0 if and only if u = 0. 29. Orthogonal unit vectors If u, and u, are orthogonal unit vec- tors and v = au, + buz, find v • 30. Cancelation in dot products In real-number multiplication, if uv = uvz and u # 0, we can cancel the u and conclude that vy = vz. Does the same rule hold for the dot product? That is, if u•v = u•v2 and u + 0, can you conclude that v = v2? Give reasons for your answer. 31. If u and v are orthogonal, show that proj, u = 0.
27. a. Cauchy-Schwartz inequality Since u · v = |u||v| cos 0, show that the inequality Ju•v| < |u||v| holds for any vectors u and v. b. Under what circumstances, if any, does |u• v| equal |u||v|? Give reasons for your answer. 28. Dot multiplication is positive definite Show that dot multipli- cation of vectors is positive definite; that is, show that u ·u 2 0 for every vector u and that u · u = 0 if and only if u = 0. 29. Orthogonal unit vectors If u, and u, are orthogonal unit vec- tors and v = au, + buz, find v • 30. Cancelation in dot products In real-number multiplication, if uv = uvz and u # 0, we can cancel the u and conclude that vy = vz. Does the same rule hold for the dot product? That is, if u•v = u•v2 and u + 0, can you conclude that v = v2? Give reasons for your answer. 31. If u and v are orthogonal, show that proj, u = 0.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:27. a. Cauchy-Schwartz inequality Since u · v = |u||v| cos 0,
show that the inequality Ju•v| < |u||v| holds for any
vectors u and v.
b. Under what circumstances, if any, does |u• v| equal |u||v|?
Give reasons for your answer.
28. Dot multiplication is positive definite Show that dot multipli-
cation of vectors is positive definite; that is, show that u ·u 2 0
for every vector u and that u · u = 0 if and only if u = 0.
29. Orthogonal unit vectors If u, and u, are orthogonal unit vec-
tors and v = au, + buz, find v •
30. Cancelation in dot products In real-number multiplication,
if uv = uvz and u # 0, we can cancel the u and conclude that
vy = vz. Does the same rule hold for the dot product? That is, if
u•v = u•v2 and u + 0, can you conclude that v = v2? Give
reasons for your answer.
31. If u and v are orthogonal, show that proj, u = 0.
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