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# 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”.

<figure><img src="/files/NzWG6jMdCfWSqONwpErl" alt=""><figcaption></figcaption></figure>

### Choosing different basis vectors

Consider a random pair of vectors, $$\vec{v}$$ and $$\vec{w}$$.&#x20;

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

The linear combination of two vectors can be used to describe ***every*** possible two-dimensional vector.&#x20;

<figure><img src="/files/rnA2tUPHozfZY5oh1IZx" alt=""><figcaption></figcaption></figure>

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 $$\hat{i},\hat{j}$$&#x20;

<figure><img src="/files/2BBXBODcHr0HRQx3qnfp" alt=""><figcaption></figcaption></figure>