What is the angle between two space diagonals of the cube ?

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

Take any corner of the cube as the origin and assign an arbitrary right-handed three-dimensional Cartesian coordinate system along the sides of the cube. By doing so, the space diagonals are reduced to two vectors , for example, $\vec{D_1}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{D_2}=\hat{i}-\hat{j}+\hat{k}$ .

To find the angle between these vectors, we take the dot product :

$\vec{D_1}\cdot \vec{D_2}=|\vec{D_1}||\vec{D_2}|\cos{\theta}$

Substituting and solving for $\theta$ , we obtain $\theta = \cos^{-1}{\dfrac{1}{3}}$