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Let $\mathbb{P}$ be a projective $n$-space. For $p=[a_0,\cdots,a_n], q=[b_0,\cdots,b_n]$ I know that the line pass through $p$ and $q$ is defined by the set $\{ [xa_0+yb_0,\cdots, xa_n+yb_n] | [x,y] \in \mathbb{P}^1\}$

I wonder the defining equation of this projective line. and can we formulate the defining equation of the line passing through two point?

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  • $\begingroup$ Dear Sang, you must not speak of "the defining equation of this projective line": the line is described by $n-1$ linear equations and I am sure someone on the site will carefully explain that to you. $\endgroup$ – Georges Elencwajg Mar 17 '13 at 11:57
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I am not sure if this is what you want. If $n=2$, you can think of the projective points $p,q$ are two vectors in the corresponding vector space. Then the projective line through these two points corresponds to the class of planes parallel to these two vectors. By taking cross product of these two vectors, you then find the normal vector of the planes, and hence you can formulate the equation. If $n>2$, the normal vector is not unique, then I am not sure how you can formulate it.

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  • $\begingroup$ Your answer is incorrect because the corresponding vector space you mention has dimension $4$, not $3$ [notice that the indices of coordinates start from $0$], and cross-product doesn't make sense there. Anyway, Sang's problem has absolutely nothing to do with cross products or normal vectors. $\endgroup$ – Georges Elencwajg Mar 17 '13 at 12:05
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The situation isn't really any different from the affine case: the equation of a line in $n$ dimensions is a full rank system of $n-1$ linear equations in $n$ variables.

Homogenizing, in the projective case it is a full rank system of $n-1$ homogeneous linear equations in $n+1$ variables.

As a quick sanity check, the solution to such a thing is two-dimensional. And we have two degrees of freedom: one to choose a point on the line, and one to rescale the coordinates.

To find the system of equations, you are asking to find the matrix $A$ of rank $n-1$ satisfying

$$A \left( \begin{matrix}a_0 & b_0 \\ a_1 & b_1 \\ \vdots & \vdots \\ a_n & b_n \end{matrix} \right) = 0$$

That is, you are simply asking to do a left null space matrix calculation.

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