This short explanation is included for the sake of those who have had a calculus course in which the geometry of the cross product has not been made clear. Unfortunately, this is the case in most of the current books. The geometric significance in terms of angles and lengths is very important, not just a formula for computing the cross product.
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 A.0.1 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,
Definition A.0.2 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.
 (1.1) 
For α a scalar,
 (1.2) 
For a,b, and c vectors, one obtains the distributive laws,
 (1.3) 
 (1.4) 
Formula 1.1 follows immediately from the definition. The vectors a × b and b × a have the same magnitude,
× a  = −a× =
− 
= b × a + c × a. 
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 A.0.3 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
a × b  =
i+
j+ 

+
k.  (1.5) 
Proof: From the above table and the properties of the cross product listed,




 (1.6) 
■
It is probably impossible for most people to remember 1.5. Fortunately, there is a somewhat easier way to remember it. Define the determinant of a 2 × 2 matrix as follows

Then
 (1.7) 
where you expand the determinant along the top row. This yields


Note that to get the scalar which multiplies i you take the determinant of what is left after deleting the first row and the first column and multiply by
 (1.8) 
which is the same as 1.6. There will be much more presented on determinants later. For now, consider this an introduction if you have not seen this topic.
Use 1.7 to compute this.

Example A.0.5 Find the area of the parallelogram determined by the vectors,

These are the same two vectors in Example A.0.4.
From Example A.0.4 and the geometric description of the cross product, the area is just the norm of the vector obtained in Example A.0.4. Thus the area is
Example A.0.6 Find the area of the triangle determined by
This triangle is obtained by connecting the three points with lines. Picking
Observation A.0.7 In general, if you have three points (vectors) in ℝ^{3},P,Q,R the area of the triangle is given by
