Phonon
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Calculate distance in 3D space
12 votes

It's Pythagorean theorem, just like with 2D space. $||[0, 0, 0]-[1, 2, 3]|| = \sqrt{(0-1)^2+(0-2)^2+(0-3)^2} = \sqrt{1+4+9} = \sqrt{14}$

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Improper integral of exponential function times the integer part of $x$
Accepted answer
3 votes

Break the integral up into a sum of integrals over intervals of length 1. This way the floor of $x$ becomes constant for that interval. You get $$\int_0^\infty = \sum_{k=0}^{\infty}k\left(\int_k^{k+1}...

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Are these two matrix products equivalent?
3 votes

Yes, this is always true... sorrrrt of, given that $a$ and $b$ are of the same length, but you must be careful! In $(b^T * a) * r$, the last multiplication is that of a scalar by a matrix (a vector ...

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how seconds per year does the average worker work?
2 votes

Assuming factory workers usually belong to unions, we should assume that they work exactly 40 hours per week. Assuming the current state of the economy, the answer in fact approaches 0, but that's for ...

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$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . what can you say about the direction of $\vec{b} \times \vec{c}$?
1 votes

Hint: The fact that $\vec{a} \times \vec{b}$ = $\vec{c} \times \vec{d}$ means that $\vec{a}$ and $\vec{b}$ lie in the same plane as $\vec{c}$ and $\vec{d}$. What does this tell you about a ...

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Two different results for splitting sample points into groups of a certain size?
0 votes

The original question was asking about How many ways 12 persons may be divided *into three groups of 4 persons each*?. That's not the same as how many possible groups of 4 people can be formed from 12....

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Nonlinear Matrix Equations
0 votes

Well, you have 3 equations and 5 unknowns. With algebraic manipulations (by substituting for $A$ and then for $B$) you can get it to the form $$-X^TX + I + XC + D = 0.$$ This way you're left with one ...

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Dissecting a square into congruent pieces that all touch the centre
0 votes

There's an infinite number of solutions. The general form involves a big + with its two lines crossing at the center of the square and perpendicular to each other. When you rotate it at any angle, it ...

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