I'm working through Spivak's Calculus textbook on Chapter 1, 19d.
The question posed is as follows:
Now the part I'm lost on is 19d. I was able to find the only when cases for parts a and b, but now I'm lost for part c. How do I find the case that equality holds only when $x_1 = (lambda)(y_1)$ and $x_2 = (lambda)(y_2)$?
A hint is fine, I'm not just looking for an answer. I'm just really stuck on this one. I have a feeling it has something to do with the terms $(x_1y_1 + x_2y_2)^2$ or $(x_2y_1 - x_1y_2)^2$ but I'm not sure.
Thank you ahead of time.
For the (c) part of 19(d), given that equality of Schwarz inequality holds, i.e.
$$\color{red}{(x_1y_1 + x_2y_2)^2} = \color{blue}{(x_1^2+x_2^2)(y_1^2+y_2^2)}$$
Using $\color{blue}{(x_1^2+x_2^2) (y_1^2+y_2^2)} = \color{red}{(x_1y_1 + x_2y_2)^2} + \color{green}{(x_1y_2 - x_2y_1)^2}$, the identity from part (c):
$$\begin{align*} \color{green}{(x_1y_2-x_2y_1)^2} &= 0\\ x_1y_2 - x_2y_1 &= 0\\ x_1y_2 &= x_2y_1\\ \end{align*}$$
Either $y_1=y_2=0$, or WLOG assume $y_2 \ne 0$,
$$\begin{align*} x_1 &=\frac{x_2y_1}{y_2}\\ x_1 &= \frac{x_2}{y_2} \cdot y_1 \end{align*}$$
By taking $\lambda = \frac{x_2}{y_2}$, there is a number $\lambda$ that satisfies both $x_1 = \lambda y_1$ and $x_2 = \lambda y_2$.