Question

Find the volume of the parallelepiped with adjacent edges pq, pr, ps. Given p(2, 0, 1), q(-3, 1, 7), r(4, 2, 0), s(0, 5, 4)?

Answer 1

To find the volume of a parallelepiped with adjacent edges pq, pr, and ps, we can calculate the cross product of any two adjacent edges. Then, using the scalar triple product, we can find the volume by taking the absolute value of the dot product of the cross product and the third edge.

Step-by-step explanation:

To find the volume of a parallelepiped, we need to first calculate the cross product of any two adjacent edges. Let's choose the edges pq and pr. The cross product of pq and pr gives us the third edge ps. Next, we can use the scalar triple product to find the volume. The scalar triple product of the vectors pq, pr, and ps is equal to the volume of the parallelepiped:

V = |pq · (pr x ps)|

Substituting the given coordinates, we have:

pq = q - p = (-3, 1, 7) - (2, 0, 1) = (-5, 1, 6)

pr = r - p = (4, 2, 0) - (2, 0, 1) = (2, 2, -1)

ps = s - p = (0, 5, 4) - (2, 0, 1) = (-2, 5, 3)

Now, we can calculate the cross product of pq and pr:

pr x pq = (2, 2, -1) x (-5, 1, 6) = (-13, -12, -12)

Taking the dot product of pq and (pr x pq):

pq · (pr x pq) = (-5, 1, 6) · (-13, -12, -12) = 65 + (-12) + 72 = 125

Therefore, the volume of the parallelepiped is:

V = |pq · (pr x pq)| = |125| = 125 units cubed.