Vector Definition: Explaining Left Side to Right Side

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Hi For some strange reason I just can't see why this is true?? Can anyone help me explain why the left side can be written as the right side? I added a picture.
 

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The thing that confuses me the most is the p-tu.
 
It's just the associative law for addition of vectors:
(v- p)+ (p- tu)= v+ (-p+ p)- tu= v- tu.
 
Ahh that's the trick huh? So is (v- p)+ (p- tu) a geometric figure? Or just something to be considered as numbers.
 

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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