Final answer:
To find the volume of the given parallelepiped, we use the formula V = |(pq × pr) · ps|. We calculate the vectors pq, pr, and ps by subtracting the coordinates of the points. Then, we find the cross product (pq × pr) and take the dot product with ps. The resulting absolute value is the volume of the parallelepiped, which is 32 cubic units.
Step-by-step explanation:
To find the volume of the parallelepiped with adjacent edges pq, pr, and ps, we can use the formula V = |(pq × pr) · ps|, where × represents the cross product and · represents the dot product. First, we find the vectors pq, pr, and ps by subtracting the coordinates of the points. Then, we calculate the cross product (pq × pr) and take the dot product with ps. Finally, we take the absolute value of the result to get the volume.
Given p(-2, 1, 0), q(3, 5, 3), r(1, 4, -1), and s(3, 6, 4), we can calculate pq = q - p = (3, 5, 3) - (-2, 1, 0) = (5, 4, 3), pr = r - p = (1, 4, -1) - (-2, 1, 0) = (3, 3, -1), and ps = s - p = (3, 6, 4) - (-2, 1, 0) = (5, 5, 4).
Next, we calculate the cross product (pq × pr) = (5, 4, 3) × (3, 3, -1) = (15, -6, -3) - (12, -15, 20) = (3, 9, -23). Finally, we take the dot product (3, 9, -23) · (5, 5, 4) = (15 + 45 - 92) = -32.
Taking the absolute value of -32, we get the volume of the parallelepiped as |(-32)| = 32 cubic units.