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Properties of Cross Product
If u, v, and w are any vectors in 3-space and k is any scalar, then
(a)
(b)
(c)
(d)
(e)
(f)
The proofs follow immediately from Formula 1 and properties of determinants; for example, (a) can be proved as follows:
Proof (a) Interchanging u and v in 1 interchanges the rows of the three determinants on the right side of 1 and hence changes the
sign of each component in the cross product. Thus .
The proofs of the remaining parts are left as exercises.
EXAMPLE 3 Standard Unit Vectors
Consider the vectors
These vectors each have length 1 and lie along the coordinate axes (Figure 3.4.1). They are called the standard unit vectors in
3-space. Every vector in 3-space is expressible in terms of i, j, and k since we can write
For example,
From 1 we obtain
