1
$\begingroup$

Let $x_1, x_2, x_2, y_1, y_2, y_3 \in \mathbb{R}^{+}$. Is it true that if $x_1^2+x_2^2<x_3^2$ and $y_1^2+y_2^2<y_3^2$ then $(x_1+y_1)^2+(x_2+y_2)^2<(x_3+y_3)^2$ ? Thanks in advance.

$\endgroup$
2
$\begingroup$

We will show that

$$ (x_1 + y_1) ^2 + (x_2 + y_2)^2 \leq \left( \sqrt{ x_1^2 + x_2^2} + \sqrt{ y_1^2 + y_2^2} \right)^2 < (x_3 + y_3)^2$$

This is equivalent to

$$ 2x_1y_1 + 2 x_2 y_2 \leq 2 \sqrt{ x_1^2 + x_2^2 } \sqrt{ y_1 ^2 + y_2^2 } $$

This is equivalent to

$$2 x_1x_2y_1y_2 \leq x_1^2y_2^2 + x_2^2 y_1^2$$

This is equivalent to

$$ 0 \leq ( x_1 y_2 - x_2y_1)^2 $$

Hence, it is true.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.