Three applets for linear algebra or multivariable calculus

This semester I’m teaching two sections of Calculus III (multivariable calculus) and I happen to be teaching the first four weeks of Linear Algebra. The first couple weeks of both courses cover properties of vectors in Rⁿ. (Of course, just to confuse the instructor and the students who happen to be in both classes, the order, emphasis, and notation is different in both textbooks!)

I created three vector-related GeoGebra applets to use in the classes. I thought others might want to see them too.

The first applet showcases all the basic vector operations: addition, scalar multiplication, subtraction, linear combinations, the dot product, vector projections, and the cross product. I’ve used this applet in class to show the students how the operations work. I’ve also given them the link so they can play with it on their own (they can even play with it in class, since we are in a computer-equipped classroom).

The other two applets are intended for the student use. The intent is for the students to discover properties of the dot product and the cross product on their own. All I told them was that the dot and cross products were two attempts to define vector-vector multiplication. Then I sent the to the computers to play with the applets. I didn’t define anything ahead of time. There are questions below the applets that guide them through the discovery process. Then we came back together to discuss their findings. I was very pleased with how it worked. For example, one group of students figured out that $\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos\theta$ .

(Since some students were in both my Calculus III and Linear Algebra classes and since we didn’t cover the cross product in Linear Algebra, I only used these applets in Calculus III.)

Dot product applet: