Find the angle between any two diagonals of a cube.

To find the angle between any two diagonals of a cube, we first need to understand the geometry of the cube.

Understanding Cube Diagonals

A cube has several diagonals. There are two types: face diagonals and space diagonals. Here, we focus on the space diagonals, which connect opposite corners of the cube.

Space Diagonal Calculation

In a cube with side length \( a \), the length of a space diagonal can be calculated using the formula:

Diagonal Length = \( a\sqrt{3} \)

Finding the Angle

To find the angle between two space diagonals, we can use the dot product of the vectors representing these diagonals. Consider the cube positioned in a 3D coordinate system:

• Diagonal 1: from (0, 0, 0) to (a, a, a)
• Diagonal 2: from (0, a, 0) to (a, 0, a)

The vectors for these diagonals are:

• Vector 1: \( \vec{d_1} = (a, a, a) \)
• Vector 2: \( \vec{d_2} = (a, -a, a) \)

The dot product of these vectors is:

\( \vec{d_1} \cdot \vec{d_2} = a^2 - a^2 + a^2 = a^2 \)

The magnitudes of the vectors are:

• Magnitude of \( \vec{d_1} = a\sqrt{3} \)
• Magnitude of \( \vec{d_2} = a\sqrt{3} \)

Using the formula for the cosine of the angle \( \theta \) between two vectors:

\( \cos(\theta) = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|} \)

Substituting the values:

\( \cos(\theta) = \frac{a^2}{(a\sqrt{3})(a\sqrt{3})} = \frac{1}{3} \)

Final Angle Calculation

To find the angle \( \theta \), take the inverse cosine:

\( \theta = \cos^{-1}\left(\frac{1}{3}\right) \)

This angle is approximately \( 70.53^\circ \). Thus, the angle between any two space diagonals of a cube is about \( 70.53^\circ \).