Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 11 - Vectors and the Geometry of Space - 11.1 Exercises - Page 756: 66

Answer

$$a = 2,{\rm{ }}b = - 3$$

Work Step by Step

$$\eqalign{ & {\rm{Let\ }}{\bf{u}} = \left\langle {1,2} \right\rangle ,{\rm{ }}{\bf{w}} = \left\langle {1, - 1} \right\rangle ,{\rm{ }}{\bf{v}} = \left\langle { - 1,7} \right\rangle \cr & {\rm{Where\ }}{\bf{v}} = a{\bf{u}} + b{\bf{w}} \cr & \left\langle { - 1,7} \right\rangle = a\left\langle {1,2} \right\rangle + b\left\langle {1, - 1} \right\rangle \cr & \left\langle { - 1,7} \right\rangle = \left\langle {a,2a} \right\rangle + \left\langle {b, - b} \right\rangle \cr & \left\langle { - 1,7} \right\rangle = \left\langle {a + b,2a - b} \right\rangle \cr & {\rm{Therefore,}} \cr & \left\{ \matrix{ a + b = - 1 \hfill \cr 2a - b = 7 \hfill \cr} \right. \cr & {\rm{Solving\ the\ system\ of\ equations}} \cr & a + b = - 1 \cr & \underline {2a - b = 7} \cr & 3a = 6 \cr & a = 2 \cr & b = - 1 - a = - 3 \cr & {\rm{Then,}} \cr & a = 2,{\rm{ }}b = - 3 \cr} $$
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