Unit Vectors / Basis Vectors

In the $xy$-coordinate system, there are two special vectors.

• The one pointing to the right with length $1$, commonly called “i hat” $\hat{i}$ or “the unit vector in the x-direction”.

• The other one is pointing straight up with length $1$, commonly called “j hat” $\hat{j}$ or “the unit vector in the y-direction”.

Choosing different basis vectors

Consider a random pair of vectors, $\vec{v}$ and $\vec{w}$.

$a\vec{v} + b\vec{w}$

The linear combination of two vectors can be used to describe every possible two-dimensional vector.

So while we can use a random pair of vectors as basis vectors, forming a new coordinate system, the association would be different from the version using the standard basis of $\hat{i},\hat{j}$