Why is a line $\mathbf{l}$ in $\mathbb{P}^2$ defined as a linear combination of its 2D null-space as $\mathbf{x} = \mu \mathbf{a} + \lambda \mathbf{b}$ where $\mathbf{l}^T\mathbf{a} = \mathbf{l}^T\mathbf{b}=0$? Specifically, I do not understand why it can be any linear combination of scalars $\mu$ and $\lambda$. I suspect it has to do with homogeneous coordinates and how scaling does not matter but I cannot wrap my head around this and have trouble coming up with a good intuition as why this is the case. I cannot visualize this but I am able to confirm it algebraically.
Algebraically, I tried to do a simple example of a line with two points $\mathbf{a} = (1,0,1)$ and $\mathbf{b} = (0,1,1)$. Such a line is simply $\mathbf{l}=\mathbf{a} \times \mathbf{b} = (-1, -1, 1)$. Suppose $\mu=2, \lambda=1$, the point we get is $\mathbf{x} = (2,1,3)$ which is a point on $\mathbf{l}$. This works for all $\mu$ and $\lambda$ which confirms the formula.