Question

The rectangular box shown in the figure has dimensions 9" × 5" × 3" (a = 9, b = 5, c = 3). Approximate the angle θ formed by a diagonal of the base and a diagonal of the 5" × 3" side. (Round your answer to two decimal places.)

Answer 1

The simplest method to do this is with the vector dot product. Let the vector A = <9i + 5j> with magnitude √106 be the base diagonal, and B = <5j + 3k> be the diagonal vector on the side, with magnitude √34. Then cos θ = (A dot B) divided by the product of the magnitudes. A dot B =30, so 

cos θ = 30 / √(34 x 106)

 = 0.4997 ==>

 θ = 49.19° is the answer