42 CHAPTER 3. VECTOR PRODUCTS

a

b

c

In this picture the vector c points upwards from the plane determined by the othertwo vectors. You should consider how a right hand system would differ from a left handsystem. Try using your left hand and you will see that the vector c would need to point inthe opposite direction as it would for a right hand system.

From now on, the vectors, i, j,k will always form a right handed system. To repeat,

k

i

j

if you extend the fingers of your right hand along i and closethem in the direction j, the thumb points in the direction ofk. The following is the geometric description of the crossproduct. It gives both the direction and the magnitude andtherefore specifies the vector.

Definition 3.4.2 Let a and b be two vectors in R3. Then a×bis defined by the following two rules.

1. |a×b|= |a| |b|sinθ where θ is the included angle.

2. a×b ·a = 0, a×b ·b = 0, and a,b,a×b forms a right hand system.

Note that |a×b| is the area of the parallelogram determined by a and b.

b

aθ

|b|sin(θ)

The cross product satisfies the following properties.

a×b =−(b×a) , a×a = 0, (3.19)

For α a scalar,(αa)×b = α (a×b) = a×(αb) , (3.20)

For a,b, and c vectors, one obtains the distributive laws,

a×(b+ c) = a×b+a× c, (3.21)

(b+ c)×a = b×a+ c×a. (3.22)

Formula 3.19 follows immediately from the definition. The vectors a×b and b×ahave the same magnitude, |a| |b|sinθ , and an application of the right hand rule shows they