/sci/ - Science & Math

>>11543665
Suppose [math]P[/math] is a projection operator onto a subspace of a vector space [math]V[/math] with inner product.
Then, naturally, [math]I-P[/math] is the projection onto the subspace orthogonal to [math]P[/math]'s image.
Finally, we obtain the "generalized Pythagoras" formula [math]\langle u, v \rangle = \langle Pu + (I-P)u, Pv+ (I-P)v \rangle = \langle Pu, Pv \rangle + \langle (I-P)u, (I-P)v \rangle[/math]. In particular, consider [math]u = (1, 1)[/math], [math]v = (1, -1)[/math] , and [math]P[/math] as projection onto the first entry.
>>11542341
>quickly
Probably not.

>>11485755
>Name a single (1) instance of a physical or chemical process where you need the concept of entropy.
Literally any kinda of design work for pumps/turbines/impellers/blowers/fans/nozzle, etc etc etc you fucking ignoramus.

>>11442462
Any element [math] a [/math] in a group has an identity element [math] a' [/math]. So [math] abc=e \implies a'abc=a'e \implies ebc=a'e \implies bc=a' \implies bca=a'a=aa'=e [/math]
>>11441512
>>11440363
It's pretty simple, just the definition of tangent. [math] \tan\alpha_1=f_c/a\ ;\ \tan\alpha_2=(f_d-f_c)/b\ ;\ \tan\alpha_3=f_d/c [/math]
[eqn] a\tan\alpha_1+b\tan\alpha_2=a\cdot\frac{f_c}{a}+b\cdot\frac{f_d-f_c}{b}=c\cdot\frac{f_d}{c}=c\tan\alpha_3 [/eqn] Genuinely don't know what the "g" is tho. Probably a typo.
>>11440771
I can't help you, but you deserve a thoughtful and informative reply for the gorgeous TeX along.