Give a point an address

One number, one axis

On a number line, the coordinate 2 means two units from the origin in the positive direction. The coordinate $-2$ lies two units in the opposite direction. The origin is the point labeled zero.

In a plane, a point needs two coordinates. With the usual horizontal and vertical axes, $(2,1)$ means “two units right, then one unit up.” The order matters: $(1,2)$ is generally a different point.

Build the point one direction at a time

The dashed path separates the coordinate changes. The straight segment connects the origin directly to the point. Both reach the same address. In 3D, a third coordinate supplies a third direction. Rotating the view changes the picture on the screen, not the point’s coordinates.

Three-dimensional drawings on a flat screen are projections. Two different points can overlap in a particular view. Rotate the view to reveal the missing separation.

A line is a collection of addresses

The equation $y=2x+1$ describes all pairs whose second coordinate is twice the first plus one. For $x=0,1,2$, the points are $(0,1),(1,3),(2,5)$. The slope is 2: increasing $x$ by one increases $y$ by two. The intercept is 1: the line crosses the vertical axis at $(0,1)$.

A vertical line such as $x=2$ is still a line, even though it cannot be written as a single-valued function $y=f(x)$.

In three dimensions,

describes a plane. Choose $x$ and $y$, then calculate $z$. Points $(0,0,0)$, $(1,0,2)$, and $(0,1,1)$ all lie on it. A plane extends indefinitely; the patch in a drawing only shows part of it.

A vector can mean a displacement

The point $(2,1)$ is an address. The vector $\begin{bmatrix}2\\1\end{bmatrix}$ can describe a displacement: move 2 in the first direction and 1 in the second. When we draw that vector starting at the origin, its tip reaches the point with the same coordinates.

For two points $(x_1,y_1)$ and $(x_2,y_2)$, their displacement is $(x_2-x_1,y_2-y_1)$. The Euclidean distance between them is

The distance from $(0,0)$ to $(3,4)$ is $\sqrt{9+16}=5$. A coordinate can be negative; distance cannot.

More coordinates than we can draw

A student might be represented by five features: hours studied, classes attended, sleep, practice problems, and quiz score. That is a point in $\mathbb R^5$. We cannot directly draw five perpendicular axes in our physical space, but we can still compute with five coordinates. High-dimensional math keeps the coordinate bookkeeping even when a complete picture is unavailable.