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Chapter 11: Problem 78

Find the angle between the diagonal of a cube and the diagonal of one of its sides.

Short Answer

Expert verified
The angle between the diagonal of a cube and the diagonal of one of its sides is 90 degrees.

Step by step solution

01

Identify the lengths of the cube and side diagonals

For a cube with side length of \(a\), the diagonal of the side of a cube, which is a right triangle, can be found using the Pythagorean theorem. It will be \(\sqrt{a^2+a^2} = a\sqrt{2}\). The cube diagonals are the hypotenuse of a right triangle where the other two sides are the cube edge and the side diagonal. So, this length is \(\sqrt{a^2+(a\sqrt{2})^2} = a\sqrt{3}\).
02

Use the dot product formula

The dot product formula allows us to find the cosine of the angle between two vectors given their lengths and the dot product of the vectors. The dot product of these vectors is zero because they are perpendicular. So the formula would be \(cos(\theta) = \frac{0}{|a\sqrt{2}| \cdot |a\sqrt{3}|}\), which simplifies to \(cos(\theta) = 0 \).
03

Find the actual angle

Using the inverse cosine function we find that \(\theta = cos^{-1}(0)\) which equals to 90 degrees.

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