Points, Directions, and Coordinate Systems

A coordinate system gives numbers a place to live.

In a two-dimensional plane, the point (3, 2) means:

move 3 units along the first axis
move 2 units along the second axis

The vector [3, 2] can describe the same location, but it can also describe a movement:

[3, 2] = 3e_1 + 2e_2

Here, $e_1$ and $e_2$ are the coordinate directions.

A point marks where you are. A direction tells how to move.

Point Versus Direction

A point answers:

where is it?

A direction answers:

how should I move?

The numbers may look the same, but the meaning changes with context.

In ML, this distinction appears often. An embedding vector can be treated as the location of a token in a learned space. A gradient vector is better read as a direction of change.

The same coordinate list can play both roles because coordinates do not carry their role alone. The sentence around the vector gives the role. "The token is at [3, 2]" describes a point. "Move by [3, 2]" describes a direction.

MATH-C04-T02-001Exercise: Move from a point

Start at p = (2, 1) and move by direction d = [3, -1].

What point do you reach?

Check the point

Compute it first, then check your number.

HintAdd coordinate by coordinate

Add the direction coordinate by coordinate.

SolutionPoint plus direction

A direction is added to the current point coordinate by coordinate:

(2,1) + [3,-1] = (5,0)

The result is a new point, not just another direction.

MATH-C04-T02-004Exercise: Use context to choose the role

Enter 1 if [2, -4] is being used as a direction here:

Update the current point
by adding [2, -4].

Your answer

Compute it first, then check your number.

HintLook for action

The words update and adding tell you the vector is being used as a movement.

SolutionDirection from context

Enter 1. The vector tells how to change the current point, so it is being used as a direction.

Same Numbers, Different Meaning

The expression [3,2] is not enough by itself to tell us the full story.

It might mean:

the point three units right and two units up
the direction that moves three units right and two units up
a learned representation with two coordinates

Context gives the coordinates their role.

MATH-C04-T02-002Exercise: Read the role

Enter 1 if this is a direction, or 2 if it is a point:

Move by [3, -1]
from the current position.

Your answer

Compute it first, then check your number.

HintLook at the verb

The word move is the clue.

SolutionDirection reading

Enter 1. The vector describes how to move, so it is being used as a direction.

Coordinate Systems Are Conventions

Coordinates are not the object itself. They are a way to describe the object.

If you rotate or rescale the axes, the coordinate numbers may change, but the underlying point can remain the same. This is why later chapters care about bases, transformations, and invariants.

MATH-C04-T02-003Exercise: Avoid over-trusting coordinates

Enter 1 if this statement is too strong:

Coordinate numbers
are the object itself,
not a description of it.

Your answer

Compute it first, then check your number.

HintCoordinates depend on axes

If the axes change, the numbers can change.

SolutionDescription, not object

Enter 1. Coordinates are a description relative to chosen axes. They are not the object itself.

Before Moving On

Read coordinates as descriptions, not as the object itself. A coordinate system tells you how to name positions and directions.