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I learned to denote lines like following, where $\vec{p}$ is direction vector, $P$ is point in the line and < > is the notation for linear span of vector(s):

$$p: X = P + <\vec{p}>$$

This is meant to be equivalent to:

$$p: X = P + t\cdot\vec{p},\ t\in R$$

Now a plane with point $P$ and vectors $\vec{p},\vec{q}$ can be written as:

$$\rho:\ X = P + t\cdot\vec{p}+u\cdot\vec{q},\ t,u\in R$$

And I would be inclined to write it as:

$$\rho:\ X = P + <\{\vec{p}, \vec{q}\}>$$

Is that correct? I have no other reasons to use linear span than laziness (no need to define $t$ and remember not to use that letter any more) and the fact that I find it more obvious.

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    Looks alright. Perhaps you could choose to simplify to $\langle \vec p, \vec q \rangle$ since you didn't put the $\vec p$ in a set (singleton) in the case of the line either. – 2017-01-31
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    For a more pedantic flavour, you could write $\{P\} + \operatorname{sp} \{ p,q \}$. – 2017-01-31

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