Visualizing the Span of Two Vectors in 3-Space

An Interactive Applet powered by Sage and MathJax.

(By Prof. Gregory V. Bard, based on a suggestion by Sepideh Stewart)

Overview

On this page, we're going to try to visualize the span of 2 vectors in ordinary 3-dimensional space.

Discussion

Now we've going to see what the span looks like for 2 particular vectors, and it turns out that it is a plane in most cases. However, if the two vectors happen to be linearly dependent then it will be a line.

The neat thing about this applet is that you can drag the picture around and view it from many different angles. Also, you can change the vectors used.

Notes

• In the above visualization, the blue plane (or sometimes a blue line) represents the span.
• The two red arrows represent the normalization of the two vectors which you have chosen. That means they are unit vectors pointing in the same direction as your vectors.
• The three green lines are the $x$, $y$, and $z$ axes.

Last modified on June 19, 2013.