Understanding Vector Direction in a Cubic System | Explanation & Examples

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What is meant by this sentence:

"In a cubic system, all directions of the vectors [100], [010], [001], [bar{1} 00], [0 bar{1} 0], and [00 bar{1}], areequivalent and indistinguishable."

They point in different directions!
 
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I assume they are talking about a crystal here. Then the statement just says that you cannot tell along which of the given vectors you are looking by examining the crystal.
 
Is plane (1 1 2) identical to (-1 -1 -2), or are they just parallel?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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