Possible explanation on infinite number of Planes, and a Line

Question

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?

Answer 1

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 :

$$\begin{pmatrix} y_v(z_B - z_A) - z_v(y_B - y_A) \\ z_v (x_B - x_A) - x_v (z_B - z_A) \\ x_v (y_B - y_A) - y_v (x_B - x_A) \end{pmatrix}$$

Answer 2

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

$$ ex - dy = ea - db. $$

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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.