3d Coordinate System

Introduction

Coordinate planes

The following illustration shows the 3d coordinate system with the three coordinate planes:

  • The \(xy\)-plane, which contains the \(x\)-axis and the \(y\)-axis, and is perpendicular to the \(z\)-axis

    The points on this plane have \(z\)-coordinate equal to 0.

  • The \(xz\)-plane, which contains the \(x\)-axis and the \(z\)-axis, and is perpendicular to the \(y\)-axis

    The points on this plane have \(y\)-coordinate equal to 0.

  • The \(yz\)-plane, which contains the \(y\)-axis and the \(z\)-axis, and is perpendicular to the \(x\)-axis

    The points on this plane have \(x\)-coordinate equal to 0.

You can interact with the illustration by turning it around and viewing it from different angles.

Projections of a point

A point \(P\) with coordinates \((x,y,z)\) in a 3d coordinate system has a number of projections.

  • First there are the projections to the coordinate planes:

    • Its projection to the \(xy\) plane is the nearest point to \(P\) that lies on the \(xy\)-plane. In other words, this is the nearest point to \(P\) with zero \(z\)-coordinate. Its coordinates are \((x,y,0)\).

    • Its projection to the \(xz\) plane is the nearest point to \(P\) that lies on the \(xz\)-plane. In other words, this is the nearest point to \(P\) with zero \(y\)-coordinate. Its coordinates are \((x,0,z)\).

    • Its projection to the \(yz\) plane is the nearest point to \(P\) that lies on the \(yz\)-plane. In other words, this is the nearest point to \(P\) with zero \(x\)-coordinate. Its coordinates are \((0,y,z)\).

  • Then there are projections of \(P\) to each of the three coordinate axes:

    • The projection to the \(x\)-axis is the point on the \(x\)-axis that is nearest to \(P\). Since it is on the \(x\)-axis, the \(y\) and \(z\) coordinates must both be 0. The coordinates of this projection are \((x,0,0)\).

    • Similarly, the projection of \(P\) to the \(y\)-axis, or the point on the \(y\)-axis that is nearest to \(P\), is \((0,y,0)\).

    • Finally, the projection of \(P\) to the \(z\)-axis, or the point on the \(z\)-axis that is nearest to \(P\), is \((0,0,z)\).

Those 6 projections, together with the point \(P\) itself and the origin, form a rectangular box with the point \(P\) and the origin at the directly opposite corners:

If \(P\) lies on one of the coordinate planes, axes, or at the origin, this box will degenerate into a rectangle, an interval, or a single point.

The following illustration combines both the “coordinate box” and the coordinate planes, so you can see the projections on the actual planes:

Try it yourself

A bit of a stretch

See if you can figure out the answers to these (Hint: the distance from a point \(P\) to a plane or a line is a distance from \(P\) to the point on the plane or line that is nearest to \(P\).)