Why is the equation of line through 2 points in projective space is their cross product

Question

Can anyone give intuition that why the equation of line passing through two points in projective space is given by their cross product?

Answer 1

The cross product of two lines gives the equation of the plane that contains them in your space $k^3$, when you goes to the projective $\mathbb P^2_k$, it means the line through both points.

Answer 2

In $\mathbb P^2$, a point $\mathbf p$ is on a line $\mathbf l$ iff $\mathbf l^T\mathbf p = 0$. If you interpret this equation in terms of inhomogeneous vectors in a three-dimensional Euclidean space, it’s the familiar condition that two vectors are orthogonal iff their Euclidean scalar product—relative to the standard basis, their dot product—vanishes. So, the vector that represents the line through two points $\mathbf p$ and $\mathbf q$ is orthogonal to their homogeneous coordinate vectors. This is exactly what the cross product gives you. You can also reach this conclusion by examining the scalar triple product and what it means for it to vanish.