parametric vector form

A solution set written as $\mathbf p + t\mathbf d$ , a base point plus a direction times a parameter.

Parametric vector form writes an infinite solution set as a single vector equation: assign each free variable a parameter (say $x_2 = t$ ), write every coordinate as a function of $t$ , and stack them, e.g. $\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \mathbf p + t\,\mathbf d$ . $\mathbf p$ is the solution at $t=0$ ; $\mathbf d$ is the direction the solution moves as $t$ ranges over $\mathbb R$ . A system with $k$ free variables needs $k$ parameters and $k$ direction vectors, added the same way.