The cross product is the other way of multiplying two vectors in ℝ^{3}. It is very different from the dot product in many ways. First the geometric meaning is discussed and then a description in terms of coordinates is given. Both descriptions of the cross product are important. The geometric description is essential in order to understand the applications to physics and geometry while the coordinate description is the only way to practically compute the cross product.
Definition 3.4.1 Three vectors a,b,c form a right handed system if when you extend the fingers of your right hand along the vector a and close them in the direction of b, the thumb points roughly in the direction of c.
For an example of a right handed system of vectors, see the following picture.
In this picture the vector c points upwards from the plane determined by the other two vectors. You should consider how a right hand system would differ from a left hand system. Try using your left hand and you will see that the vector c would need to point in the 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, if you extend the fingers of our right hand along i and close them in the direction j, the thumb points in the direction of k.
The following is the geometric description of the cross product. It gives both the direction and the magnitude and therefore specifies the vector.
Definition 3.4.2 Let a and b be two vectors in ℝ^{3}. Then a × b is defined by the following two rules.
Note that
The cross product satisfies the following properties.
 (3.22) 
For α a scalar,
 (3.23) 
For a,b, and c vectors, one obtains the distributive laws,
 (3.24) 
 (3.25) 
Formula 3.22 follows immediately from the definition. The vectors a × b and b × a have the same magnitude,

A proof of the distributive law is given in a later section for those who are interested.
Now from the definition of the cross product,

With this information, the following gives the coordinate description of the cross product.
Proposition 3.4.3 Let a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k be two vectors. Then
 (3.26) 
Proof: From the above table and the properties of the cross product listed,




 (3.27) 
■
It is probably impossible for most people to remember 3.26. Fortunately, there is a somewhat easier way to remember it.
 (3.28) 
where you expand the determinant along the top row. This yields
 (3.29) 
which is the same as 3.27. Determinants are discussed in Volume 2 but some people may have seen them. All you need here is how to evaluate 2 × 2 and 3 × 3 determinants.

and

Here is the rule: You look at an entry in the top row and cross out the row and column which contain that entry. If the entry is in the i^{th} column, you multiply
Use 3.28 to compute this.
=
i−
j+
k 

= 3i + 5j + k. 
Example 3.4.5 Find the area of the parallelogram determined by the vectors

These are the same two vectors in Example 3.4.4.
From Example 3.4.4 and the geometric description of the cross product, the area is just the norm of the vector obtained in Example 3.4.4. Thus the area is
Example 3.4.6 Find the area of the triangle determined by
This triangle is obtained by connecting the three points with lines. Picking
Observation 3.4.7 In general, if you have three points (vectors) in ℝ^{3},P,Q,R the area of the triangle is given by
