Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 5 - Inner Product Spaces - 5.2 Inner Product Spaces - 5.2 Exercises - Page 247: 98


$$\langle u, v \rangle= \frac{1}{4}u_1v_1+\frac{1}{9}u_2v_2.$$

Work Step by Step

The figure shows an ellipse centered at the origin and its equation is given by $$\frac{x^2}{2^2}+\frac{y^2}{3^2}=1 \quad \quad(1)$$ Now, since $\|u\|=1$, then $$\|u\|^2 = \langle u, u \rangle= c_1u_1^2+c_2u_2^2, \quad \quad (2).$$ Comparing (1) and (2), we get $$c_1=\frac{1}{4}, \quad c_2=\frac{1}{9}.$$ Consequently, the inner product is given by $$\langle u, v \rangle= \frac{1}{4}u_1v_1+\frac{1}{9}u_2v_2.$$
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