Possible explanation on infinite number of Planes, and a Line If I have a line L with its vectored form being stated as:
$$<x,y,z> = <a,b,c> + t<d,e,f>$$
or in another way as this:
\begin{align}
x &= a+dt \\
y &= b+et \\
z &= c+ft
\end{align}
If I want a plane that contains that line I have an infinite number of planes right? Recognizing this conundrum how can I establish a rapport of what is need to find a single plane that the line resides in?
 A: Take any vector not colinear to $(d, e, f)$, say $(g, h, i)$. Then you get a plane parametrized by :
$$\begin{eqnarray} x = a + d t + gu\\
y = b + et + hu \\
z = c + ft + iu
\end{eqnarray} \quad t, u \in \mathbb{R}$$
But having the parametrization of a plane is unpractical. The best way to describe a plane is as follow :
Take a non-zero vector $\vec{n}$ normal to the plane, and a point $A$ belonging to the plane, then a point $B$ belongs to this plane $\Leftrightarrow \vec{AB}\cdot \vec{n} = 0$, where $\cdot$ is the scalar product. Say $\vec{n}$ has coordinates $(x_n, y_n, z_n)$, we get that $B$ belongs to that plane iff 
$$(x_B - x_A) x_n + (y_B - y_A) y_n + (z_B - z_A) z_n = 0 $$
Let $B$ be a point outside our line. We can find the equation of the plane containing our line and the point $B$ as follow :
A normal vector is given by the cross product $\vec{v} \times \vec{AB}$. The coordinates of this normal vector are :
$$\begin{pmatrix} y_v(z_B - z_A) - z_v(y_B - y_A) \\
z_v (x_B - x_A) - x_v (z_B - z_A) \\
x_v (y_B - y_A) - y_v (x_B - x_A)
 \end{pmatrix}$$ 
A: One way to do it is to put the line in the form:
$$ t = \frac{x - a}{d} = \frac{y - b}{e} = \frac{z - c}{f}. $$
If you select any two of these, you get a plane containing your line. For instance with the first two:
$$ \frac{x - a}{d} = \frac{y - b}{e} $$
yields the plane
$$ ex - dy = ea - db. $$

In general, if you take two of these planes, e.g.
\begin{align*}
ex - dy &= ea - db \\
fx - dz &= fa - dc
\end{align*}
you can take a linear combination of the two to get all such planes:
$$ \lambda(ex - dy) + \mu(fx - dz) = \lambda(ea - db) + \mu(fa - dc). $$
For all $\lambda$ and $\mu$ as long as at least one of $\lambda$ and $\mu$ is non-zero.
