Definition A.1.1 A parallelepiped determined by the three vectors, a,b, and c consists of
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That is, if you pick three numbers, r,s, and t each in
The following is a picture of such a thing.You notice the area of the base of the parallelepiped,
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This expression is known as the box product and is sometimes written as
Example A.1.2 Find the volume of the parallelepiped determined by the vectors, i + 2j − 5k,i + 3j − 6k,3i + 2j + 3k.
According to the above discussion, pick any two of these, take the cross product and then take the dot product of this with the third of these vectors. The result will be either the desired volume or minus the desired volume.
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Now take the dot product of this vector with the third which yields
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This shows the volume of this parallelepiped is 14 cubic units.
There is a fundamental observation which comes directly from the geometric definitions of the cross product and the dot product.
Proof: This follows from observing that either
Notation A.1.4 The box product a × b ⋅ c = a ⋅ b × c is denoted more compactly as