The set of Vectors that defines space is called the basis.
We refer to these vectors as basis vectors.
In 2d space, basis vectors are commonly defined as and .
These particular vectors are called the standard basis vectors. We can think of them as 1 in the direction of X and 1 in the direction of Y.
We can think of all other vectors in the space as Linear Combinations of the basis vectors.
For example, if I have vector , we can treat each component as scalar (see Vector Scaling) for the basis vectors: .
We can choose any set of vectors as the basis vectors for space, giving us entirely new coordinate systems. However, they must meet the following criteria:
- They're linear independent. That means you cannot get one Vector by just scaling the other.
- They span the space. That means, by taking a linear combination of the two scaled vectors, you can return any vector.
Basis vectors don't have to be orthogonal to each other, but transformations become more challenging with a non-orthogonal basis.
In applications that require visualizing 3d space, like a game studio or a 3d modeling program, it's common for the basis vectors of the space to be displayed prominently. They are typically colored , the color order mapping each axis to a color in RGB according to this post.
Basis vectors in PlayCanvas
3Blue1Brown (2016) Dye et al. (2018)
David Dye, Sam Cooper, and Freddie Page. Mathematics for Machine Learning: Linear Algebra - Home. 2018. URL: https://www.coursera.org/learn/linear-algebra-machine-learning/home/welcome. ↩
3Blue1Brown. Linear combinations, span, and basis vectors. August 2016. URL: https://www.youtube.com/watch?v=k7RM-ot2NWY. ↩