The cross product is a special way to multiply two vectors in three-dimensional space.
The cross product is linked inextricably to the determinant, so we will first introduce the determinant before introducing this new operation.
Typically, when one computes the determinant of a matrix, we think of the terms as follows.
Determinants in have many uses. In , one of the uses is the definition of the cross product.
is efficient, because you immediately have the components of the desired vector without additional simplification.
Notice that the cross product is not commutative! In fact, it is anticommutative, meaning the statement below.
Let’s examine the cross product on famous unit vectors.
One way to remember the cross products of the unit vectors , , and is to use the diagram below.
Since we see that the cross product of two basic unit vectors produces a vector orthogonal to both unit vectors, we are led to our next theorem (which could be verified through brute force computations).
In three-dimensional space, when seeking a vector perpendicular to both and , we could choose one of two directions: the direction of , or the direction of . The direction of the cross product is given by the right-hand rule. Given and in with the same initial point, point the index finger of your right hand in the direction of and let your middle finger point in the direction of (much as we did when establishing the right-hand rule for the 3-dimensional coordinate system). Your thumb will naturally extend in the direction of . If you switch your fingers, pointing the index finder in the direction of and the middle finger in the direction of , your thumb will now point in the opposite direction, allowing you to “visualize” the anticommutative property of the cross product.
Just as we related the angle between two vectors and their dot product, there is a similar relationship relating the cross product of two vectors to the angle between them. Before we get started, we need an identity.
The theorems above help us make a strong connection between the cross product and geometry.
Note that if and are in , we can still use the cross product to compute the area of the parallelogram spanned by and . We just add a -component of to each vector.
In addition to the geometric applications we have already seen, we can also use the cross product in some physical applications.
Imagine turning a wrench. The wrench originates at a point and terminates at a point . Let . You apply a force to the end of the wrench. If points in the same direction as , the bolt will not twist at all, since you will just be pulling on the handle. If is perpendicular to the handle, then we expect quite a bit of twisting to occur.
When a charged particle moves through a magnetic field, it experiences a force. If the charge is , the velocity of the particle is , and the magnetic field is , then the force is given by
Below, we summarize some rules for working with cross products.
- Respects scalar multiplication:
- Relation to parallel vectors:
- , , and .
Moreover, these properties determine the cross product uniquely.
We will not prove that the cross product is the only function with these properties, but that is an important point. If you ever wondered where this crazy formula came from, the uniqueness of the cross product is your explanation. If you want these properties, there is only one operation which gives them to you, and it is the cross product. We leave you with the following curious fact. The cross product only exists in and . While a proof of this fact is beyond the scope of this course, we hope that this mystery encourages you to travel deeper into your studies.
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