I need to calculate the volume of a parallelepiped using the scalar triple product. I can do this if i have vectors. however instead we've been given 8 points which mark the corners. HELPPP this is the question:
Calculate the volume of the parallelepiped with vertices at:
(9,4,3), (7,12,2), (12,5,6), (3,-4,0), (10,13,5), (1,4,-1), (4,5,2) and (6,-3,3)
Duplicate post; already answered
To calculate the volume of a parallelepiped using the scalar triple product, you need to find the vectors corresponding to the three edges that meet at one of the vertices. Once you have these vectors, you can calculate their scalar triple product to determine the volume.
Let's take the first three points as an example: (9,4,3), (7,12,2), and (12,5,6).
To find the vector from the first point to the second point, subtract the coordinates of the first point from the coordinates of the second point:
Vector AB = (7-9, 12-4, 2-3) = (-2, 8, -1)
Similarly, find the vector AC by subtracting the coordinates of the first point from the coordinates of the third point:
Vector AC = (12-9, 5-4, 6-3) = (3, 1, 3)
Now, calculate the vector triple product (also known as the scalar triple product) of these vectors by taking the dot product of the vector AB with the cross product of AB and AC:
Scalar Triple Product = AB · (AB × AC)
To calculate AB × AC, take the cross product of AB and AC:
AB × AC = (-2, 8, -1) × (3, 1, 3)
To find the cross product, use the determinant method:
AB × AC = (8 * 3 - (-1) * 1, -2 * 3 - (-1) * 3, -2 * 1 - 8 * 3)
= (25, -9, -26)
Now, take the dot product of AB with the result of the cross product:
Scalar Triple Product = (-2, 8, -1) · (25, -9, -26)
Dot product = (-2 * 25) + (8 * -9) + (-1 * -26)
= -50 - 72 + 26
= -96
The volume of the parallelepiped is the absolute value of the scalar triple product:
Volume = |Scalar Triple Product| = |-96| = 96 cubic units
Repeat this process with the other three sets of points to find their respective volumes. Sum up all four volumes to get the total volume of the parallelepiped.