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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?

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    $\begingroup$ $\sqrt{1^2+1^2} = \sqrt{{1\over100}(10^2)+{1\over100}(10^2)}$ so it should work! $\endgroup$ – danimal Jun 19 '15 at 11:08
  • $\begingroup$ @danimal Ugh, I'm a moron. Still don't know why this isn't working on my vector magnitude... $\endgroup$ – Jonathan Mee Jun 19 '15 at 11:15
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When you plug them back, they are equal:

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

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

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