True or False In Exercises 83 and 84, determine

Chapter 5, Problem 84

(choose chapter or problem)

(a) The vectors \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) in \(R^{3}\) have equal lengths but opposite directions.

(b) If \(\mathbf{u}\) and \(\mathbf{v}\) are two nonzero vectors in \(R^{3}\), then \(\mathbf{u}\) and \(\mathbf{v}\) are parallel if and only if \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\).

(c) A special type of least squares approximation, the Fourier approximation, is spanned by the basis \(S=\{1, \cos x, \cos 2 x, \ldots, \cos n x, \sin x, \sin 2 x, . ., \sin n x\}\).

Text Transcription:

u X v

v X u

R^3

u

v

u X v = 0

S = {1, cos x, cos 2x, ..., cos nx, sin x, sin 2x, . ., sin nx}