Question

The standard unit hypercube in Rⁿ is a solid whose vertices are vectors with coordinates in the set {0,1}.

Calculate the angle between the side of the hypercube and the longest diagonal in the hypercube, for n=2,3,4 (you can use calculator).

Answer 1

To calculate the angle between the side of the hypercube and the longest diagonal in the hypercube, we can use the dot product formula. For n=2, the angle is approximately 45°. For n=3, the angle is approximately 54.7°. And for n=4, the angle is approximately 60°.

Step-by-step explanation:

To calculate the angle between the side of the hypercube and the longest diagonal in the hypercube, we can use the dot product formula. Let's start with n=2:

The side length of the hypercube is 1, so the side vector is [1, 0]. The longest diagonal vector is [1, 1]. Using the dot product formula, we can calculate the angle:

angle = arccos((1 * 1 + 0 * 1) / (sqrt(1^2 + 0^2) * sqrt(1^2 + 1^2))) = arccos(1 / sqrt(2)) ≈ 45°

Similarly, for n=3, the side vector is [1, 0, 0] and the longest diagonal vector is [1, 1, 1]. The angle can be calculated as:

angle = arccos((1 * 1 + 0 * 1 + 0 * 1) / (sqrt(1^2 + 0^2 + 0^2) * sqrt(1^2 + 1^2 + 1^2))) = arccos(1 / sqrt(3)) ≈ 54.7°

For n=4, the side vector is [1, 0, 0, 0] and the longest diagonal vector is [1, 1, 1, 1]. The angle calculation is:

angle = arccos((1 * 1 + 0 * 1 + 0 * 1 + 0 * 1) / (sqrt(1^2 + 0^2 + 0^2 + 0^2) * sqrt(1^2 + 1^2 + 1^2 + 1^2))) = arccos(1 / sqrt(4)) = arccos(1 / 2) ≈ 60°