(b) Show that u, v, and w are coplanar, using the definition that there exist two nonzero real numbers a and ẞ such that w = au + ẞv. Consider the plane formed by u and v in standard position. Then au + ẞv, is the vector that represents the diagonal of the parallelogram formed by au and ẞv. If nonzero real numbers α and ẞ exist such that w = au + ẞv, then u, v, and w coplanar. So we must show nonzero real numbers & and ß | exist such that w = au + ẞv. Thus, equating components in (0, -9, 18) a(1, 4, −7)+ ß(2, −1, 4), we want to solve the following system. = = a + 2ẞ 4a β = -7α + 4ẞ The solution to this system is (a, ẞ) v, and w (If an answer does not exist, enter DNE.) Thus, nonzero real numbers α and ẞ exist such that w = au + ẞv, so vectors u, coplanar.
(b) Show that u, v, and w are coplanar, using the definition that there exist two nonzero real numbers a and ẞ such that w = au + ẞv. Consider the plane formed by u and v in standard position. Then au + ẞv, is the vector that represents the diagonal of the parallelogram formed by au and ẞv. If nonzero real numbers α and ẞ exist such that w = au + ẞv, then u, v, and w coplanar. So we must show nonzero real numbers & and ß | exist such that w = au + ẞv. Thus, equating components in (0, -9, 18) a(1, 4, −7)+ ß(2, −1, 4), we want to solve the following system. = = a + 2ẞ 4a β = -7α + 4ẞ The solution to this system is (a, ẞ) v, and w (If an answer does not exist, enter DNE.) Thus, nonzero real numbers α and ẞ exist such that w = au + ẞv, so vectors u, coplanar.
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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Question
Transcribed Image Text:(b) Show that u, v, and w are coplanar, using the definition that there exist two nonzero real numbers a and ẞ such that w = au + ẞv.
Consider the plane formed by u and v in standard position. Then au + ẞv, is the vector that represents the diagonal of the parallelogram formed by au and ẞv. If nonzero real numbers α and ẞ exist such
that w = au + ẞv, then u, v, and w
coplanar. So we must show nonzero real numbers & and ß |
exist such that w = au + ẞv. Thus, equating components in
(0, -9, 18) a(1, 4, −7)+ ß(2, −1, 4), we want to solve the following system.
=
= a + 2ẞ
4a
β
= -7α + 4ẞ
The solution to this system is (a, ẞ)
v, and w
(If an answer does not exist, enter DNE.) Thus, nonzero real numbers α and ẞ
exist such that w = au + ẞv, so vectors u,
coplanar.
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