# Meaning of points and lines in a space

Say I have a Line $P$ that cuts across 2 points $(0, 1, -1)$ and $(1,-1,0)$ in a space of $\mathbb{R}^3$.

If those were 2 vectors, I could say they span a plane, 3 vectors then the entire space. Since they are points and not vectors, what does a line across these points in $\mathbb{R}^3$ mean? I cannot say that it spans a line in the row/column space because this is a line.

I tried to find out the equations to this line by doing this: $$\begin{bmatrix} 0 & 1 & -1 & 1\\ 1 & -1 & 0 & 1 \end{bmatrix}\begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}$$ $$\begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix}= t\begin{bmatrix} 2\\ 1\\ 0\\ -1 \end{bmatrix} + k\begin{bmatrix} -1\\ 0\\ 1\\ 1 \end{bmatrix}, \; \; k,t \in \mathbb{R}$$ Then I get 2 equations from the above: $$2x+y=1$$ $$-x+z=-1$$

Do these 2 equations represent the line $P$ that cut across the points $(0, 1, -1)$ and $(1,-1,0)$? But how do they represent because each of the 2 equations themselves is a line on their own.

I could continue to derive from the 2 equations to find out the solution set to $x$, $y$ and $z$: $$\begin{bmatrix} 2 & 1 & 0\\ -1 &0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} 1\\ -1 \end{bmatrix}$$ $$\begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix} + c\begin{bmatrix} 1\\ -2\\ 1 \end{bmatrix}$$

Again, at this point, I am also very confuse what this set of solution is representing. I know that this set of solution is for the 2 equations above but does it mean that if in the column space, by moving in any amount of $\begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix} + c\begin{bmatrix} 1\\ -2\\ 1 \end{bmatrix}$ would let me reach the line $P$?

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In three dimensions, a line is uniquely determined by specifying any two planes whose intersection is the line in question... – J. M. Sep 10 '11 at 16:25

Start by translating the line back to the origin and write it as $y = cx$
Is it a must to go through the implicit form of the points to find out the equations before getting the explicit form of each of the $x$, $y$ and $z$? Can I get from the points to the explicit form without having to find out the implicit form first? – xenon Sep 10 '11 at 18:08
By implicit form, I mean like expressing the line in terms of one or more equations for instance: $\left \{ \begin{bmatrix} x\\ y\\ z \end{bmatrix} \mid x+y+z=0 \; and \; x-y+2z=1\right \}$ and explicit I mean expressing the the line by having the variables $x,y,z$ as the subject like $\left \{ \begin{bmatrix} \frac{1}{2}-\frac{3}{2}t\\ -\frac{1}{2}+\frac{1}{2}t\\ t \end{bmatrix} \mid t \in \mathbb{R} \right \}$. So I was thinking if I could get to the explicit form directly without going through the implicit form first? – xenon Sep 11 '11 at 3:45