?Consider vectors \(\mathbf{u}=\langle 1,4,-7\rangle\) , \(\mathbf{v}=\langle

Chapter 2, Problem 224

(choose chapter or problem)

Consider vectors \(\mathbf{u}=\langle 1,4,-7\rangle\) , \(\mathbf{v}=\langle 2,-1,4\rangle\) , \(\mathbf{w}=\langle 0,-9,18\rangle\) , and \(\mathbf{p}=\langle 0,-9,17\rangle\) .

a. Show that u, v, and w are coplanar by using their triple scalar product

b. Show that u, v, and w are coplanar, using the definition that there exist two nonzero real numbers \(\alpha\) and \(\beta\) such that \(\mathbf{w}=\alpha \mathbf{u}+\beta \mathbf{v}\).

c. Show that u, v, and p are linearly independent—that is, none of the vectors is a linear combination of the other two.

Text Transcription:

u = langle 1,4,-7 rangle

v = langle 2,-1,4 rangle

w = langle 0,-9,18 rangle

p = langle 0,-9,17 rangle

alpha

beta

w = alpha u + beta v