Problem:

Consider the vector

$\displaystyle \vec{v}=\begin{bmatrix}1\\ 2\\ 3\\ 4\end{bmatrix}$ in $\displaystyle \mathbb{R}^{4}$

Find a basis of the subspace of R^4 consisting of all vectors perpendicular to $\displaystyle \vec{v}$.

I'm not very sure about how to solve this problem. I assume that I need to find a set of vectors $\displaystyle \vec{u_{1}},...,\vec{u_{m}}$ for which the case $\displaystyle \vec{v} \cdot \vec{u_{i}} = 0$ holds.

I tried using the formula,

$\displaystyle \vec{v}^\perp=\vec{v}-\vec{v}^\parallel$

Since I already know $\displaystyle \vec{v}$, I would just need to figure out $\displaystyle \vec{v}^\parallel$. To do this, I used projection.

$\displaystyle proj_x(\vec{v}) = \sum_{i=1}^{m}(\vec{u}_{i} \cdot \vec{v})\vec{u}_{i}$

This is where I get stuck, because I'm LOOKING for the u vectors... correct?

Am I far off the path I should be on here or am I on the right track?

Any help is greatly appreciated. Thanks for reading.