Improper Square Root Simplification

Question

I'm trying to simplify a ratio to modify a vector by. Basically I want to find a constant such that the xy-components of two vectors are equal: https://math.stackexchange.com/a/1330263/194115

So I do this: $$\sqrt{x_1^2 + x_2^2} = \sqrt{cy_1^2 + cy_2^2}$$ So I simplify as follows:

1. $\sqrt{x_1^2 + x_2^2} = \sqrt{c(y_1^2 + y_2^2)}$
2. $\sqrt{x_1^2 + x_2^2} = \sqrt c\sqrt{y_1^2 + y_2^2}$
3. $\frac{\sqrt{x_1^2 + x_2^2}}{\sqrt{y_1^2 + y_2^2}} = \sqrt c$

And by squaring both sides I get: $$\frac{x_1^2 + x_2^2}{y_1^2 + y_2^2} = c$$

But let's test this with values:

• $x_1 = 1$
• $x_2 = 1$
• $y_1 = 10$
• $y_2 = 10$

So $\frac{1}{100} = c$. But if I plug these values back into the original equation the square roots are not equal! How is this possible? What am I doing wrong here?

Answer 1

When you plug them back, they are equal:

$\sqrt{1^2+1^2}=\sqrt{2}=\sqrt{\frac{10^2}{100}+\frac{10^2}{100}}$

Answer 2

$$\sqrt{x_1^2 + x_2^2} = \sqrt{cy_1^2 + cy_2^2}$$ $$\sqrt{1^2 + 1^2} = \sqrt{c10^2 + c10^2}$$ $$\sqrt{2} = \sqrt{200c}$$ when $c=1/100$ $$\sqrt{2} = \sqrt{2}$$