The cross product is an operation that acts on vectors in three dimensions and results in another vector in three dimensions. In contrast to the dot product, the cross product is restricted to vectors in three dimensions. Consider three-dimensional vectors and and let be the angle between and . The geometric interpretation of the cross product is a vector that is perpendicular to both and (using the right hand rule), and has norm .
The algebraic interpretation of the cross product is a vector obtained by the formula:
This formula can be expressed as the determinant of a matrix:
Now, we can use various properties about determinants to prove properties of the cross product. For example, we can verify that the cross product is distributive over addition, i.e.,
The cross product satisfies the following properties:
Since is an odd function, we have .
What happens if ? We can no longer determine the unique direction of a vector that is perpendicular to (remember that we are in three dimensions), and the geometric interpretation seems unclear. However, has norm and thus .
If , what can we say about and ? By definition, implies . Therefore,
1. Show that is the positive area of the parallelogram having sides and .
Solution: We know the area of the triangle bounded by vectors and is given by . Hence, the area of the parallelogram is , which is the norm of , i.e. .
It is interesting that while area is a two-dimensional property, this uses a three-dimensional calculation.
2. Show that the cancellation law doesn't hold. If and , what can we say about the vectors ?
We have that . From the remark, since , we either have or is parallel to . In the first case, we get , and in the second case, we get for some value .
3. Prove Lagrange's Identity.
Solution. This follows immediately, since , , and .
4. Show the equivalence of the geometric interpretation and the algebraic interpretation of the cross product.
We need to show that the geometric and algebraic definitions give vectors with the same magnitude and direction. To check direction, we will show that both vectors are perpendicular to and . This is given in the definition of the geometric interpretation. For the algebraic interpretation, taking the dot product gives
Furthermore, the vectors point in the same direction since the determinant of
is positive by expanding along the third row; hence the vectors are determined by the right hand rule.
Next, we check that the vectors have the same length, by calculating the square of the norm. We have
For the algebraic interpretation, we have
By comparing terms, we see that the lengths of the vectors are equal.
The cross product has numerous applications in physics, such as describing moments, angular momentum, torque, etc.