Why is the cross product of two points a line and the cross product of two lines a point?

Question

I am currently studying projective geometry. I have trouble understanding the point-line duality concept.

Why is the cross product of two points a line and the cross product of two lines a point?

Answer 1

I'm confused as to how you want the duality here.

Let $V$ be the underlying vector space, let $k$ ($=\mathbb{R}$ or $\mathbb{C}$) be the field over which the vector space is taken (so $V=k^3$). Let $(x_1:x_2:x_3)$ be homogeneous coordinates in the projective plane $\mathbb{P}^2$ (defined by the standard basis of $V$).

Let $p_1=(\alpha_1:\alpha_2:\alpha_2)$ and $p_2=(\beta_1:\beta_2:\beta_3)$ be two points in $\mathbb{P}^2$, and let $l_1$ and $l_2$ be the corresponding lines in the underlying vector space $V$. Then $\overline{p_1p_2}$ is a line in $\mathbb{P}^2$, which is a plane in $V$; the normal to this plane is given by $l_1\times l_2=:l_3$, and the equation of the line in $\mathbb{P}^2$ is given by $(l_3)_1x_1+(l_3)_1x_2+(l_3)_3x_3=0$, where $(l_3)_i$ is the $i$-th component of $l_3$, $1\le i\le 3$.

Let $l_1$ and $l_2$ be two lines in $\mathbb{P}^2$; there are numbers $\gamma_1,\gamma_2,\gamma_3,\delta_1,\delta_2,\delta_3\in k$ such that \begin{align*} l_1&=\{(x_1:x_2:x_3)\in\mathbb{P}^2:\gamma_1x_1+\gamma_2x_2+\gamma_3x_3=0\}, \\ l_2&=\{(x_1:x_2:x_3)\in\mathbb{P}^2:\delta_1x_1+\delta_2x_2+\delta_3x_3=0\}, \end{align*} so $\gamma:=(\gamma_1,\gamma_2,\gamma_3)$ and $\delta:=(\delta_1,\delta_2,\delta_3)$ are the corresponding normals to the planes $L_1$ and $L_2$ to which $l_1$ and $l_2$ correspond in $V$. The cross product of $l_1$ and $l_2$ is the simply the cross product of $\gamma$ and $\delta$, which is a vector normal to both $\gamma$ and $\delta$; this vector necessarily lies in both planes and thus equals the intersection $L_1\cap L_2$, which is projectively the intersection $l_1\cap l_2$, and is a point in $\mathbb{P}^2$.