Cartesian line equation

The cartesian form of a 2D line is the form you are used to seeing.

It describes how the y value changes as x changes (with an offset, c):

y=mx+c

Why Cartesian form works

This equation is true, because any (x,y) point along the line will satisfy the equation.

For example, take the line equation y=4x+3, which, drawing the graph, we can see has a point at, for example, (2, 11):

• Substitute the values into the equation:
  • 11=4(2)+3
  • 11=8+3
  • 11=11 // true

While this may seem obvious, it shows us something very important:

• A line equation simply represents a ‘condition’ where, when a point is substituted into the line equation, the point is only on the line if the equation is satisfied (true).

Different form 2D line equations

You may have also seen an equation that looks like this:

x=7

All that this means is that, for any point, it is on the line if its x-coordinate is equal to 7. That means that we will get a straight vertical line, where the x-coordinate of all the points on the line is 7.

3D cartesian line equations

We don’t usually work with 3D lines in their Cartesian format (we usually convert them to vector form) - but we can do.

The same rule applies as with 2D lines - the equation of a line simply defines the conditions that a point must satisfy to be on the line.

Example: line where x=5 and z=-2

For the line with that equation, it essentially means that every point on the line has x coordinate 5 and z coordinate of -2, but the y coordinate could be anything.

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