I was told it was $90$ degrees, but then others say it is about $35.26$ degrees. Now I am unsure which one it is.
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It depends on what you mean by the skew diagonal.
Consider the cube with corners at $(x,y,z)$ where each element is either zero or one.
In particular, one diagonal is $0=(0,0,0)$ to $u=(1,1,1)$.
Now it depends on what you mean by a "skew diagonal."
If $0$ to $v=(1,1,0)$ is the other diagonal you are looking form, then the cosine of the angle is $$\frac{u\cdot v}{|u||v|}=\frac{2}{\sqrt{3}\sqrt{2}}$$
The inverse cosine of that value is approximately $35.26$ degrees.
On the other hand, if you mean a skew diagonal such as the diagonal from $(1,0,0)$ and $(0,1,0)$, then that vector of that diagonal is $w=(-1,1,0)$, and $u\cdot w=0$ so the two angles are perpendicular.
Thomas Andrews
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It depends on what you mean by the skew diagonal, though. Editing to reflect that... – Thomas Andrews Mar 17 '13 at 21:58
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Check the picture out I put. The darker lines are what I am trying to find the angle of. – Roger Mar 17 '13 at 22:08
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Yes, that is the latter situation - the 90 degree situation. You can see it directly through symmetry that it has to be 90 degrees I think. – Thomas Andrews Mar 17 '13 at 22:09
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Okay thanks. I am trying to figure the reasoning out now. The other case seemed easier to reason out. – Roger Mar 17 '13 at 22:11
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Take your cube
$\hspace{100pt}$
and make three more copies arranged like this:
$\hspace{50pt}$
Observe, that two red and two blue edges form a parallelogram, but it is symmetric (find the appropriate symmetry yourself) and as such it is a rectangle, hence, the angle between red and blue edge is $90^\circ$.
I hope this helps ;-)
dtldarek
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If we assume the cube has unit side length and lies in the first octant with faces parallel to the coordinate planes and one vertex at the origin, then the the vector $(1,1,0)$ describes a diagonal of a face, and the vector $(1,1,1)$ describes the skew diagonal.
The angle between two vectors $u$ and $v$ is given by:
$$\cos(\theta)=\frac{u\cdot v}{|u||v|}$$
In our case, we have
$$\cos(\theta)=\frac{2}{\sqrt{6}}\quad\Longrightarrow\quad\theta\approx 35.26$$
Jared
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the answer will be 90 one vector is (1,1,1) and other is (1,0,-1)
rishabh
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Welcome to Maths.SE, and thank you for your answer! It is much more likely that it will be useful for others if you include some explanation for your assertions -- I suggest you [edit] your answer to do so. – Lord_Farin Oct 29 '13 at 09:49