Dot product discovery applet

We denote the dot product of two vectors u and v in Rⁿ as u•v. Use the applet below to discover the basic properties of the dot product. The questions below the applet are intended to guide your discovery. Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Discovery questions:

• Is u•v a vector or a scalar?
• When is u•v positive, negative, and zero?
• What happens to u•v when u or v gets longer?
• Is the dot product commutative? That is, is u•v=v•u?
• What is u•u?

Check the box labeled "Show the unit circle."

• Fix one vector on the unit circle. Move the other vector around the circle and observe u•v. u•v can be expressed as a familiar, elementary function. What do you think it is?
• Is this still true if u and v aren't unit vectors?

Created by David Richeson using GeoGebra.