Elementary Linear Algebra 7th Edition

$$\langle u, v \rangle= \frac{1}{25}u_1v_1+\frac{1}{4}u_2v_2.$$
The figure shows an ellipse centered at the origin and its equation is given by $$\frac{x^2}{5^2}+\frac{y^2}{2^2}=1 \quad \quad(1)$$ 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 $$c_1=\frac{1}{25}, \quad c_2=\frac{1}{4}.$$ Consequently, the inner product is given by $$\langle u, v \rangle= \frac{1}{25}u_1v_1+\frac{1}{4}u_2v_2.$$