Geometric meaning of line equation in homogeneous coordinate In Euclidean space, a line's equation is $$ax + by + c = 0.$$ While in homogeneous coordinates,it can be represented with $$\begin{pmatrix}x &y &1\end{pmatrix}\begin{pmatrix}a\\ b\\ c\end{pmatrix} = 0.$$ I think the meaning of the homogeneous representation is that if a point is on the line, then the inner product of two vectors goes to $0$, $$X = \begin{pmatrix}x_1\\ y_1\\ 1\end{pmatrix}, V = \begin{pmatrix}a\\ b\\ c\end{pmatrix},$$ then the meaning of line equation is that $X$'s projection onto $V$ is $0$, which means $X\perp V$. Is that right?
But my intuitive understanding is that, if a point $X$ is on a line $V$, then the projection of $X$ onto $V$ should not be $0$.

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*No. There is no notion of $\perp$ between a line and a point.

*No. Here the inner product is not a projection, but a measure ($\propto$) of the minimum distance between the objects. When coincident the minimum distance is zero.

