79. Show that if a, b are nonzero vectors such that a 1 b, then there exists a vector X such that a x X = b 15 Hint: Show that if X is orthogonal to b and is not a multiple of a, then a x X is a multiple of b.
79. Show that if a, b are nonzero vectors such that a 1 b, then there exists a vector X such that a x X = b 15 Hint: Show that if X is orthogonal to b and is not a multiple of a, then a x X is a multiple of b.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
79
Transcribed Image Text:77.
Let a, b, c be nonzero vectors. Assume that b and C are not parallel, and set
V3ах (b х с),
w = (a· c) b – (a · b) c
a. Prove that:
i. V lies in the plane spanned by b and C.
ii. V is orthogonal to a.
b. Prove that W also satisfies (i) and (ii). Conclude that V and W are parallel.
c. Show algebraically that V = W (Figure 23).
bxc
ax (b× c)
Rogawski et al., Multivariable Calculus,
4e, © 2019 W. H. Freeman and Company
FIGURE 23
78. Use Exercise 77 to prove the identity
(а х b) хс — ах (bxc) — (а b)с — (b:с) а
79. Show that if a, b
are nonzero vectors such that a 1 D, then there exists a vector
X
such that
a x X = b
15
Hint: Show that if X is orthogonal to b and is not a multiple of a, then a × X is a multiple of b.
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