The Dot Product (Calculus)?

Question

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

Answer 1

Use a unit cube in the first octant.

A vector for a diagonal of a face would be A = < 1, 0, 1 >

A vector for a diagonal of the cube would be B = < 1, 1, 1 >

magnitude A = l A l = sqrt(2)

magnitude B = l B l = sqrt(3)

A(dot)B = (1*1) + (1*0) + (1*1) = 2

Rearranging the formula for the dot product,

cos(theta) = [ A(dot)B ]  / [ l A l * l B l ] = 2 / sqrt(6)

theta = arccos( 2 / sqrt(6) ) ≈ 0.615 radians ≈ 35.3 degrees.