In the context of this slide, there is no origin. This slide simply describes the line equation in vector form. (Perhaps you can expand on your question?)
the
I was just thinking that the angle between the two vectors had to be a right angle if the dot product was zero.
mpotoole
Ah! I see.
Yes, you're right that the dot product is zero if two vectors are pointing in orthogonal directions. The confusion seems to be how you're interpreting the vector $l$ geometrically. Note that $l$ and $x$ are 3x1 homogeneous vectors representing a 2D line and a 2D point, respectively.
Let's review this slide. Take the line equation at the top of the slide: $ax + by + c = 0$. Think of this as a scaled version of the double intercept form of the line equation, covered here. The line intersects the y-axis at $x = -\frac{c}{a}$, and the line intersects the x-axis at $y = -\frac{c}{b}$.
Another way to write this equation is $ax + by + c = \mathbf{x}^T \mathbf{l}$. Geometrically, the first two components $(a,b)$ represent a 2D vector that is perpendicular to your 2D line. The third component, $c$, is related to the distance of that line from the origin. If your line passed through the origin, then $c = 0$ and the inner product $\mathbf{x}^T \mathbf{l} = a x + b y + c = 0$.
Where is the origin in this case?
In the context of this slide, there is no origin. This slide simply describes the line equation in vector form. (Perhaps you can expand on your question?)
I was just thinking that the angle between the two vectors had to be a right angle if the dot product was zero.
Ah! I see.
Yes, you're right that the dot product is zero if two vectors are pointing in orthogonal directions. The confusion seems to be how you're interpreting the vector $l$ geometrically. Note that $l$ and $x$ are 3x1 homogeneous vectors representing a 2D line and a 2D point, respectively.
Let's review this slide. Take the line equation at the top of the slide: $ax + by + c = 0$. Think of this as a scaled version of the double intercept form of the line equation, covered here. The line intersects the y-axis at $x = -\frac{c}{a}$, and the line intersects the x-axis at $y = -\frac{c}{b}$.
Another way to write this equation is $ax + by + c = \mathbf{x}^T \mathbf{l}$. Geometrically, the first two components $(a,b)$ represent a 2D vector that is perpendicular to your 2D line. The third component, $c$, is related to the distance of that line from the origin. If your line passed through the origin, then $c = 0$ and the inner product $\mathbf{x}^T \mathbf{l} = a x + b y + c = 0$.
I am enlightened