Question

Find the volume of the parallelepiped with the given vertices. $$ \begin{array}{l} (0,0,0),(3,0,0),(0,5,1),(3,5,1) \\ (2,0,5),(5,0,5),(2,5,6),(5,5,6) \end{array} $$

Short answer

The volume of the parallelepiped is 75 cubic units.

Step-by-step solution

Determine the Vectors

Identify three vectors from the vertices which represent the edges of the parallelepiped. Given vertices are at (0,0,0),(3,0,0),(0,5,1) and (2,0,5). Use subtracting method to form vectors. \n A vector from (0,0,0) to (3,0,0) is \( \mathbf{a} = (3 - 0, 0-0, 0-0) = (3,0,0) \). \n A vector from (0,0,0) to (0,5,1) is \( \mathbf{b} = (0 - 0, 5-0, 1-0) = (0,5,1) \). \n A vector from (0,0,0) to (2,0,5) is \( \mathbf{c} = (2-0 , 0-0, 5-0) = (2,0,5) \).

Calculate the Cross Product

Calculate the cross product \( \mathbf{b} \times \mathbf{c} \). The cross product is given by this formula: \[ \mathbf{v} \times \mathbf{w} = (v_2w_3 - v_3w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2w_1) \] So, \( \mathbf{b} \times \mathbf{c} = (0,5,1) \times (2,0,5) = (5*5 - 1*0, 1*2 - 0*2, 0*0 - 5*2) = (25, 2, -10)

Calculate the Scalar Triple Product

Calculate the scalar triple product of \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \). This is given as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). So, the scalar triple product is \( (3,0,0) \cdot (25, 2, -10) = 3*25 + 0*2 + 0*(-10) = 75.

Determine the Volume

The volume of the parallelepiped is given by the absolute value of the scalar triple product. Because the scalar triple product is already positive, the volume is 75 cubic units.