What is the volume of a parallelepiped whose sides are given by the vectors a 3i plus wj plus k b -i plus 3j and c 2i plus 2j plus 5k?

The volume ( V ) of a parallelepiped defined by vectors a, b, and c can be calculated using the scalar triple product, given by ( V = | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) | ). Here, a = ( 3\mathbf{i} + \mathbf{j} + \mathbf{k} ), b = ( -\mathbf{i} + 3\mathbf{j} ), and c = ( 2\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} ). First, compute the cross product b × c, then take the dot product with a, and finally find the absolute value to get the volume. Calculating this yields ( V = 20 ) cubic units.