A First Course in Linear Algebra, 2017a

4.9. The Cross Product 185
Example 4.56: Volume of a Parallelepiped
Find the volume of the parallelepiped determined by the vectors
⎡ ⎤ ⎡ ⎤ ⎡
1
1 3
⃗u = ⎣ 2 ⎦,⃗v = ⎣ 3 ⎦,⃗w = ⎣ 2
−5 −6 3
⎤
⎦
Solution. According to the above discussion, pick any two of these vectors, 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
−1 times the desired volume. Therefore by taking the absolute value of the result, we obtain the volume.
We will take the cross product of ⃗u and⃗v. This is given by
=
∣
⎡
⃗u ×⃗v = ⎣
⃗i ⃗j ⃗ k
1 2 −5
1 3 −6
1
2
−5
⎤
⎡
⎦ × ⎣
1
3
−6
⎤
⎦
⎡
∣ = 3 ⃗i +⃗j + ⃗ k = ⎣
Now take the dot product of this vector with ⃗w which yields
⎡ ⎤ ⎡ ⎤
3 3
(⃗u ×⃗v) •⃗w = ⎣ 1 ⎦ • ⎣ 2 ⎦
1 3
(
= 3⃗i +⃗j + ⃗ k
= 9 + 2 + 3
= 14
This shows the volume of this parallelepiped is 14 cubic units.
)
3
1
1
⎤
⎦
(
• 3⃗i + 2⃗j + 3 ⃗ )
k
There is a fundamental observation which comes directly from the geometric definitions of the cross
product and the dot product.
Proposition 4.57: Order of the Product
Let ⃗u,⃗v, and⃗w be vectors. Then (⃗u ×⃗v) •⃗w =⃗u • (⃗v × ⃗w).
♠
Proof. This follows from observing that either (⃗u ×⃗v) • ⃗w and ⃗u • (⃗v × ⃗w) both give the volume of the
parallelepiped or they both give −1 times the volume.
♠
Recall that we can express the cross product as the determinant of a particular matrix. It turns out
that the same can be done for the box product. Suppose you have three vectors, ⃗u = [ a b c ] T ,⃗v =
[ ] T [ ] T d e f ,and⃗w = g h i . Then the box product ⃗u • (⃗v ×⃗w) is given by the following.
⎡ ⎤ ∣ ∣
⃗u • (⃗v × ⃗w) =
⎣
a
b
c
⎦ •
∣
⃗i ⃗j ⃗ k
d e f
g h i
∣