I need to use the scalar triple product. but i don't have any vectors? how am i supposed to know what the vectors all coming from one corner are??
Q: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)
Multiple post; already answered.
To calculate the volume of a parallelepiped using the scalar triple product, you need three vectors that share a common vertex. In this case, you are given eight vertices, so it is possible to find three vectors that share a common vertex.
To find three vectors sharing a common vertex, you can select any three points from the given vertices and use them as the initial and final points to form the vectors. Let's choose the vectors starting from the first vertex at (9, 4, 3). We can choose two other vertices and form two vectors. Let's choose the second vertex at (7, 12, 2) and the third vertex at (12, 5, 6).
Now we have three vectors:
Vector A: (7, 12, 2) - (9, 4, 3) = (-2, 8, -1)
Vector B: (12, 5, 6) - (9, 4, 3) = (3, 1, 3)
Vector C: (3, -4, 0) - (9, 4, 3) = (-6, -8, -3)
Next, we need to calculate the scalar triple product of these three vectors. The scalar triple product is calculated as the dot product of the cross product of any two vectors with the third vector:
Scalar Triple Product (STP) = Vector A · (Vector B × Vector C)
To find the cross product of Vector B and Vector C, we can use the determinant method:
Vector B × Vector C = |i j k |
|3 1 3 |
|-6 -8 -3|
Expanding the determinant, we get:
Vector B × Vector C = ((1 * -3) - (3 * -8))i - ((3 * -3) - (-6 * -8))j + ((3 * -8) - (1 * -6))k
= (-3 + 24)i - (9 - 48)j + (-24 + 6)k
= 21i + 39j - 18k
Now, we can calculate the dot product of Vector A with the cross product of Vector B and Vector C:
Scalar Triple Product (STP) = Vector A · (Vector B × Vector C)
= Vector A · (21i + 39j - 18k)
= (-2, 8, -1) · (21, 39, -18)
= (-2 * 21) + (8 * 39) + (-1 * -18)
= -42 + 312 + 18
= 288
The value of the scalar triple product is 288.
Finally, we can calculate the volume of the parallelepiped using the formula:
Volume = |Scalar Triple Product (STP)|
Therefore, the volume of the parallelepiped is:
Volume = |288|
= 288
Hence, the volume of the parallelepiped with the given vertices is 288 cubic units.