[ [ rectangular parallelepipeds]

The edges of a rectangular parallelepiped are equal to $a, b$, and $c$. Find the angle between the intersecting diagonals of two faces sharing the common edge $a$.

#
Let $A B C D A 1 B 1 C 1 D 1$ be a rectangular parallelepiped, in which $A A 1\|B B 1\| C C 1\| D D 1, A B=a, A D=b$, $A A 1=c$. We need to find the angle between the diagonal $A 1 C 1$ of the face $A 1 B 1 C 1 D 1$ and the diagonal $A B 1$ of the face $A A 1 B 1 B$. Since $A 1 C 1 \| A C$, the angle between the skew lines $A 1 C 1$ and $A B 1$ is equal to the angle between the intersecting lines $A C$ and $A B 1$. We draw a section of the parallelepiped by a plane passing through the vertices $A, B 1$, and $C$. We obtain a triangle with sides

$$
A C=\sqrt{a^{2}+b^{2}}, A B 1=\sqrt{a^{2}+c^{2}}, B 1 C=\sqrt{b^{2}+c^{2}}
$$

By the cosine theorem, we find that

$$
\begin{gathered}
\cos \angle C A B 1=\frac{A C^{2}+A B^{2}-C B_{1}^{2}}{2 A C \cdot A B_{1}}= \\
=\frac{a^{2}+b^{2}+a^{2}+c^{2}-b^{2}-c^{2}}{2 \sqrt{\left(a^{2}+b^{2}\right)\left(a^{2}+c^{2}\right)}}=\frac{a^{2}}{\sqrt{\left(a^{2}+b^{2}\right)\left(a^{2}+c^{2}\right)}}>0
\end{gathered}
$$

Therefore, the angle between the lines $A 1 C 1$ and $A B 1$ is

$$
\arccos \frac{a^{2}}{\sqrt{\left(a^{2}+b^{2}\right)\left(a^{2}+c^{2}\right)}}
$$

| Let $A B C D A 1 B 1 C 1 D 1$ be a rectangular parallelepiped, in which $A A 1\|B B 1\|C C 1\| D D 1, A B=a, A D=b$, $A A 1=$. Denote $\overrightarrow{A B}=\vec{a}, \overrightarrow{A D}=\vec{b}, \overrightarrow{A A_{1}}=\vec{c}$. Then

$$
\overrightarrow{A_{1} C_{1}}=\vec{a}+\vec{b}, \overrightarrow{A B_{1}}=\vec{a}-\vec{c}
$$

If $\gamma$ is the angle between the lines $A 1 C 1$ and $A B 1$, then

$$
\cos \gamma=\frac{\left|\overrightarrow{A_{1} \vec{l}^{2}} \cdot \overrightarrow{A_{1}}\right|}{A_{1} C_{1} \cdot A B_{1}}=\frac{|(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{c})|}{\sqrt{a^{2}+b^{2}} \cdot \sqrt{a^{2}+c^{2}}}=\frac{\vec{a}^{2}}{\sqrt{a^{2}+b^{2}} \cdot \sqrt{a^{2}+c^{2}}}>0
$$

Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_6bf60d3f257a54792c0cg-27.jpg?height=105&width=428&top_left_y=425&top_left_x=815)

## Answer

$\arccos \frac{a^{2}}{\sqrt{\left(a^{2}+b^{2}\right)\left(a^{2}+c^{2}\right)}}$.