Michael Corral: Vector Calculus

28 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE
We defined the determinant as a scalar, derived from algebraic operations on scalar
entries in a matrix. However, if we put three vectors in the first row of a 3×3 matrix,
then the definition still makes sense, since we would be performing scalar multiplication
on those three vectors (they would be multiplied by the 2×2 scalar determinants
as before). This gives us a determinant that is now a vector, and lets us write the
cross product of v=v 1 i+v 2 j+v 3 k and w=w 1 i+w 2 j+w 3 k as a determinant:
∣ i j k ∣∣∣∣∣∣∣∣ ∣
v×w=
v 1 v 2 v 3 =
v 2 v ∣∣∣∣∣ ∣
3
i −
v 1 v ∣∣∣∣∣ ∣
3
∣w ∣w 1 w 2 w
2 w j+
v 1 v ∣∣∣∣∣
2
3 ∣w 1 w k
3 ∣w 1 w 2
3
= (v 2 w 3 −v 3 w 2 )i+(v 3 w 1 −v 1 w 3 )j+(v 1 w 2 −v 2 w 1 )k
Example 1.16. Let v=4i−j+3k and w=i+2k. Then
i j k
v×w=
4 −1 3
=
−1 3
∣ ∣∣∣∣∣ ∣ 1 0 2
∣ 0 2
∣ i − 4 3
∣ ∣∣∣∣∣ 1 2
∣ j + 4 −1
1 0
∣
∣ k=−2i−5j+k
The scalar triple product can also be written as a determinant. In fact, by Example
1.12, the following theorem provides an alternate definition of the determinant of a
3×3 matrix as the volume (or negative volume) of a parallelepiped whose adjacent
sides are the rows of the matrix. The proof is left as an exercise for the reader.
Theorem 1.17. For any vectors u=(u 1 ,u 2 ,u 3 ), v=(v 1 ,v 2 ,v 3 ), w=(w 1 ,w 2 ,w 3 ) in 3 :
∣ u 1 u 2 u ∣∣∣∣∣∣∣∣
3
u·(v×w)=
v 1 v 2 v 3
(1.15)
∣w 1 w 2 w 3
Example 1.17. Find the volume of the parallelepiped with adjacent sides u=(2,1,3),
v=(−1,3,2), w=(1,1,−2) (see Figure 1.4.9).
Solution: By Theorem 1.15, the volume of the parallelepiped
P is the absolute value of the scalar triple product of the three
adjacent sides (in any order). By Theorem 1.17,
u·(v×w)=
∣
= 2
∣
2 1 3
−1 3 2
1 1 −2
3 2
1 −2
∣
− 1
∣
∣
−1 2
1 −2
+ 3
∣
∣
= 2(−8)−1(0)+3(−4)=−28, so
vol(P)=|−28|=28.
−1 3
1 1
∣
x
u
0
w
z
v
Figure 1.4.9 P
y