A cube is placed so that one corner is at the origin and three edges are along the ?x -, y -, and ?z -axes of a coordinate system (Fig. P1.80). Use vectors to compute (a) the angle be-tween the edge along the ?z -axis (line ?ab?) and the diagonal from the origin to the opposite corner (line ?ad?), and (b) the angle between line a? c? (the diagonal of a face) and line ?ad?.

Solution 91P Step 1: Suppose, the side of the cube as “a” units. Then, the magnitude of vector ab = a The side bc = cd = a Since the face bcda is a square, the length bd will be, 2 2 2 2 bd = c + cd = a +a = 2a Step 2: According to the law of vector addition, ad = ab + bd The magnitude of vector ad will be, 2 2 2 2 ad = ab + bd + (2 × ab × bd × cos ) = + 2a = 3a = 90° since ab and bd are perpendicular. Cos 90 = 0