Michael Corral: Vector Calculus

24 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE
The following theorem summarizes the basic properties of the cross product.
Theorem 1.14. For any vectors u, v, w in 3 , and scalar k, we have
(a) v×w=−w×v
Anticommutative Law
(b) u×(v+w)=u×v+u×w
(c) (u+v)×w=u×w+v×w
(d) (kv)×w=v×(kw)=k(v×w)
(e) v×0=0=0×v
(f) v×v=0
(g) v×w=0if and only if v‖w
Distributive Law
Distributive Law
Associative Law
Proof: The proofs of properties (b)-(f) are straightforward. We will prove parts (a)
and (g) and leave the rest to the reader as exercises.
(a) By the definition of the cross product and scalar multiplication,
we have:
v
z
v×w
v×w=(v 2 w 3 −v 3 w 2 ,v 3 w 1 −v 1 w 3 ,v 1 w 2 −v 2 w 1 )
=−(v 3 w 2 −v 2 w 3 ,v 1 w 3 −v 3 w 1 ,v 2 w 1 −v 1 w 2 )
=−(w 2 v 3 −w 3 v 2 ,w 3 v 1 −w 1 v 3 ,w 1 v 2 −w 2 v 1 )
w
0
y
=−w×v
x
w×v
Note that this says that v×w and w×v have the same
magnitude but opposite direction (see Figure 1.4.6).
Figure 1.4.6
(g)Ifeithervorwis0thenv×w=0bypart(e),andeitherv=0=0worw=0=0v,
so v and w are scalar multiples, i.e. they are parallel.
If both v and w are nonzero, andθis the angle between them, then by formula
(1.11), v×w=0if and only if‖v‖‖w‖ sinθ=0, which is true if and only if sinθ=0
(since‖v‖>0and‖w‖>0). So since 0 ◦ ≤θ≤180 ◦ , then sinθ=0if and only ifθ=0 ◦
or 180 ◦ . But the angle between v and w is 0 ◦ or 180 ◦ if and only if v‖w. QED
Example 1.11. Adding to Example 1.7, we have
i×j=k j×k=i k×i=j
j×i=−k k×j=−i i×k=−j
i×i=j×j=k×k=0
Recall from geometry that a parallelepiped is a 3-dimensional solid with 6 faces, all
of which are parallelograms. 6
6 Anequivalentdefinitionofaparallelepipedis: thecollectionofallscalarcombinationsk 1 v 1 +k 2 v 2 +k 3 v 3
of some vectors v 1 , v 2 , v 3 in 3 , where 0≤k 1 ,k 2 ,k 3 ≤ 1.