# 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. Updated by Ryan G. Hornberger.)

## 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 8th, 2022.