# Standard basis

There is another way of thinking about Euclidean vectors. It might just seem like an alternative notation, but it’s a bit more than that. It works like this: there are only three vectors (or two in two-space) that you need to describe any other vector. These are called the standard basis vectors:

$\displaystyle \hat{i} = \left [ 1 , 0 , 0 \right ] , \quad \hat{j} = \left [ 0 , 1 , 0 \right ] , \quad \hat{k} = \left [ 0 , 0 , 1 \right ]$.

Say we have the vector $\displaystyle \vec{v} = \left [ 4 , - 2 , 7 \right ]$, which is in component form. How would we express it in standard basis? Like this:

$\displaystyle \vec{v} = 4 \hat{i} - 2 \hat{j} + 7 \hat{k}$.

This way of describing vectors is useful when manipulating and simplifying vector expressions because you can just treat them as ordinary variables. You can use your algebra skills—combining like terms and so on—just as you always have. Say that, in addition to $\displaystyle \vec{v}$, we have $\displaystyle \vec{u} = 5 \hat{j} - 2 \hat{k}$. Let’s simplify the expression $\displaystyle 3 \vec{v} + 4 \vec{u}$:

$\displaystyle 3 \vec{v} + 4 \vec{u} = 3 \left ( 4 \hat{i} - 2 \hat{j} + 7 \hat{k} \right ) + 4 \left ( 5 \hat{j} - 2 \hat{k} \right ) = 12 \hat{i} + 14 \hat{j} + 13 \hat{k}$.