# 5: Graphical Explorations of Vectors

- Page ID
- 63401

We already looked at the basics of graphing vectors. In this chapter, we’ll explore these ideas more fully. One often gains a better understanding of a concept by “seeing” it. For instance, one can study the function \(f(x) = x^{2}\) and describe many properties of how the output relates to the input without producing a graph, but the graph can quickly bring meaning and insight to equations and formulae. Not only that, but the study of graphs of functions is in itself a wonderful mathematical world, worthy of exploration.

We’ve studied the graphing of vectors; in this chapter we’ll take this a step further and study some fantastic graphical properties of vectors and matrix arithmetic. We mentioned earlier that these concepts form the basis of computer graphics; in this chapter, we’ll see even better how that is true.

- 5.1: Transformations of the Cartesian Plane
- Previously, we limited our visual understanding of matrix multiplication to graphing a vector, multiplying it by a matrix, then graphing the resulting vector. In this section we’ll explore these multiplication ideas in greater depth. Instead of multiplying individual vectors by a matrix A, we’ll study what happens when we multiply every vector in the Cartesian plans by A.

- 5.3: Visualizing Vectors- Vectors in Three Dimensions
- We ended the last section by stating we could extend the ideas of drawing 2D vectors to drawing 3D vectors. Once we understand how to properly draw these vectors, addition and subtraction is relatively easy. We’ll also discuss how to find the length of a vector in 3D.

Thumbnail: A linear combination of one basis set of vectors (purple) obtains new vectors (red). If they are linearly independent, these form a new basis set. The linear combinations relating the first set to the other extend to a linear transformation, called the change of basis. (CC0; Maschen via Wikipedia)