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)
A parallelipiped has six faces, consisting of three pairs of parallelograms across from and parallel to each other. Only four points are necessary to define such a figure; you have eight. Four vertices must be overdetermined. Knowing which four to use is the hard part of this problem. Look for sets of connecting lines that are parallel, and choose three nonparallel lines that connect to one point.
Define the vectors leading from one vertex point to three adjacent points. The vector from (9,4,3) to (7,12,2) would be -2i + 8j -1k, for example.
Then use scalar triple product rule as shown at
http://answerboard.cramster.com/calculus-topic-5-300804-0.aspx
To calculate the volume of a parallelepiped using the scalar triple product, you need to have vectors representing the three edges that meet at a common corner of the parallelepiped. In this case, you are given eight vertices representing corners of the parallelepiped.
To find the vectors, you can choose any three corners of the parallelepiped as long as they share a common corner. For example, let's choose the corners (9,4,3), (7,12,2), and (12,5,6). We can denote these corners as points A, B, and C, respectively.
To find the vector AB, which represents the edge from A to B, you subtract the coordinates of A from the coordinates of B:
AB = (7,12,2) - (9,4,3) = (-2,8,-1)
Similarly, you can find the vectors for the other two edges:
AC = (12,5,6) - (9,4,3) = (3,1,3)
AD = (3,-4,0) - (9,4,3) = (-6,-8,-3)
Now that you have vectors representing three edges of the parallelepiped, you can calculate the scalar triple product.
The scalar triple product of three vectors (a, b, c) is calculated using the equation:
a · (b × c)
where "·" denotes the dot product and "×" denotes the cross product.
In this case, the scalar triple product is:
V = AB · (AC × AD)
To calculate this, first find the cross product of AC and AD:
AC × AD = (3,1,3) × (-6,-8,-3)
To calculate the cross product, you can use the determinant method:
(3,1,3) × (-6,-8,-3) = (1*(-3) - 3*(-8), 3*(-3) - 3*(-6), 3*(-8) - 1*(-6))
= (-24, -3, -18)
Now, calculate the dot product of AB with the cross product (AC × AD):
AB · (AC × AD) = (-2,8,-1) · (-24, -3, -18)
To calculate the dot product, multiply corresponding components of the vectors and sum them:
(-2,8,-1) · (-24, -3, -18) = (-2*(-24) + 8*(-3) + (-1)*(-18))
= (48 - 24 + 18)
= 42
Finally, the volume of the parallelepiped can be calculated as the absolute value of this scalar triple product, divided by 6:
Volume = |AB · (AC × AD)| / 6
= |42| / 6
= 7
Therefore, the volume of the parallelepiped is 7 cubic units.