Question

Consider trying to determine the angle between an edge of a cube and its diagonal (a line joining opposite vertices through the center of the cube). a) Draw a large sketch of the problem and label any relevant parts of your sketch. (Hint: it will simplify things if your edges are of length one, one corner of your cube is at the origin, and your edge and diagonal emanate from the origin) b) Determine the angle between an edge of a cube and its diagonal (use arccosine to represent your answer).

Answer 1

• Answer:

The angle between an edge of a cube and its diagonal is:

θ = arccos 1/√3

Explanation:

Theta Symbol: (θ), Square-root Symbol: (√):

• Set up the problem: Let the Cube have Side Lengths of 1, Place the cube so that One Corner is at the Origin (0, 0, 0), and the Edge and Diagonal emanate from the origin.

• Identify relevant points:

Label the Points:

A(0, 0, 0)

B(1, 0, 0)

C(1, 1, 1)

• Where A is the Origin:

AB is the Edge

AC is the Diagonal

• Calculate the lengths of the Edge and Diagonal:

The Lenth of the Edge AB is (1) Since it's the side length of the cube.

• The length of the Diagonal AC can be found using the Distance Formula:

AC = √(1 - 0)^2 + (1 - 0)^2 + (1 - 0)^2 = √3

• Use the product formula:

The Dot Product Formula:

u * v = |u| |v| cos θ, Where θ is the angle between the vectors:

• Calculate the Dot Product of AB and AC:

AB = (1, 0, 0 )

AC = (1, 1, 1 )

AB * AC = (1)(1) + (0)(1) + (0)(1) = 1

• Substitute the Lengths and Dot Product into the formula:

1 = (1)(√3) cos θ

• Solve for the angle (θ):

Divide both sides by √3

cos θ = 1/√3

• Take the arccosine of both sides:

θ = arccos 1/√3

• Draw the conclusion:

Therefore, The angle between an edge of a cube and its diagonal is:

θ = arccos 1/√3

I hope this helps!