Math Insight

Applet: The dot product as projection

The dot product of the vectors $\vc{a}$ (in blue) and $\vc{b}$ (in green), when divided by the magnitude of $\vc{b}$, is the projection of $\vc{a}$ onto $\vc{b}$. This projection is illustrated by the red line segment from the tail of $\vc{b}$ to the projection of the head of $\vc{a}$ on $\vc{b}$. You can change the vectors $\vc{a}$ and $\vc{b}$ by dragging the points at their ends or dragging the vectors themselves. Notice how the dot product is positive for acute angles and negative for obtuse angles. The reported number does not depend on $\|\vc{b}\|$ only because we've divided through by that magnitude.

Applet file: dot_product_projection.ggb

Applet links

This applet is found in the pages

List of all applets

General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.

You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.