The Cross Product
Lesson 4 of 9
Objective: SWBAT find the cross product of two vectors.
Today's lesson will revolve around the cross product of two vectors and its geometric implications. Like yesterday's dot product lesson, we are learning tools that will allow us to find the equation of a plane.
It can be difficult to make a lesson like this meaningful to students since we are learning a complicated formula and not giving an application until tomorrow. In my Selling a Lesson reflection, I talk about how I combat this concept that may seem unnecessary to students.
I begin the lesson by explaining that we are going to be learning a new operation called the cross product. I say that it is an operation just like yesterday's dot product, but that the answer has a geometric meaning. Here are some introductory points that I will make:
- When you take the cross product of two vectors (notation: u x v) your answer will be a vector, not a scalar like with the cross product.
- The answer to the cross product has an important geometric property - it is orthogonal (perpendicular) to the two vectors you started with. This image can help students to see what is happening geometrically.
- I ask students if there could be a perpendicular vector to any two vectors. I also ask if there could be more than one orthogonal vector to a given pair of vectors. This will help establish that the cross product is not commutative and you will get different vectors depending on the order.
- I remind students that yesterday we said that if two vectors are perpendicular, their dot product must be zero. That fact will be important in the derivation of the cross product formula since we are thinking about orthogonal vectors.
Once we get some of the introductory information taken care of, we can dive right in to the cross product formula. The YouTube video below is a nice resource to show your students to build upon yesterday's work with the dot product. The derivation of the cross product formula is lengthy, and I would never expect my students to be able to replicate it, but exposing them to some of the pertinent ideas will make it more meaningful to them. Thus, you may choose to show all or some of this video, but at least students will loosely understand where it comes from.
After the video we will go through an example together of finding the cross product of two vectors. The method I use involves matrices and cofactors. I have students set up a matrix like this and then make new 2x2 matrices and find the determinant of each to find the i, j, and k coefficients of the cross product vector. I instruct students that to find the 2x2 matrix for the i coefficient, you delete the i row and column of the 3x3 matrix and then take the determinant of this new 2x2 matrix. The process is repeated for the j and k coefficients. Furthermore, the coefficients of the i, j, and k terms will alternate positive and negative signs.
After going through one example together, students are usually in a good place to try finding the cross product on their own. I will put two vectors u and v on the board and have one half of the class find u x v and the other half find v x u. After working, students will share their answers and we will see that something has happened - the coefficients of the cross product vectors are opposites. Thus the cross product operation is not commutative. I will ask students to think about this and usually a student will realize that the new vectors are both orthogonal to u and v, but they are going in different directions.
To end this lesson, I will ask the following questions to recap the big ideas of what we learned today.
- How is the cross product vector geometrically related to u and v?
- How are the vectors u x v and v x u related?
- What does orthogonal mean?
- What vector operation was essential to derive the cross product formula?
After these concluding questions, I will assign some problems from our textbook as homework.