I have always understood vectors to be sets of numbers such that their components are coefficients in the general form of a linear equation. For example, take the vector $(2, 1)$. I would draw this in 2-dimensional space like so:
In other words, the first component is the coefficient for x, and the second component is the coefficient for y. This generalizes to n-dimensional space.
Where this falls apart is when I try to compute the slope of this line. Since I assume we are in general form:
$$ Ax + By + C = 0 $$ $$ 2x + y + 0 = 0 $$ $$ y = -2x $$
But that means that the slope of the line $m = -2$, which is:
I've tried this a few times, and it seems like the vector actually represents a line that is orthogonal to the one I had in mind. I feel like I'm going crazy or missing something really basic. What is wrong with my understanding?
Let $\mathbf v=(A,B)$ be your vector. The equation $Ax+By+C=0$ is equivalent to $\mathbf v\cdot(\mathbf x-\mathbf p)=0$, which says that $\mathbf v$ is orthogonal to the line, not parallel to it, just as you discovered. If you instead consider the line given parametrically by $\mathbf x=t\mathbf v+\mathbf p$, the slope will be $B/A$ as you expected, since this line is parallel to the vector.