Find the angle between a diagonal of a cube and an adjacent edge. (Enter your answer in degrees. Round your answer to two decimal places.)

To find the angle between a diagonal of a cube and an adjacent edge, we can use the concept of vectors and dot product.

1. Consider a cube with side length 'a'.

2. The diagonal of the cube can be represented as a vector d = ai + aj + ak. The magnitude of this vector (length of the diagonal) is sqrt(a^2 + a^2 + a^2) = a*sqrt(3).

3. An edge of the cube can be represented as a vector e = ai. The magnitude of this vector (length of the edge) is a.

4. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. So, we have d.e = |d|*|e|*cos(theta), where theta is the angle we want to find.

5. The dot product d.e = a^2 + 0 + 0 = a^2.

6. Substituting the values into the dot product equation, we get a^2 = a*sqrt(3)acos(theta), which simplifies to cos(theta) = 1/sqrt(3).

7. Therefore, theta = arccos(1/sqrt(3)).

8. Converting this to degrees (since 1 radian = 180/pi degrees), we get theta = arccos(1/sqrt(3)) * (180/pi) = 54.74 degrees.

So, the angle between a diagonal of a cube and an adjacent edge is approximately 54.74 degrees.