Locating a point in space requires three mutually perpendicular coordinate axes — the \(x\)-, \(y\)- and \(z\)-axes — meeting at the origin \(O\). These determine three coordinate planes (the \(XY\), \(YZ\) and \(ZX\) planes), which divide space into eight octants. A point \(P\) is given by an ordered triplet \((x,y,z)\), where \(x,y,z\) are its perpendicular distances from the \(YZ\), \(ZX\) and \(XY\) planes respectively; there is a one-to-one correspondence between points and triplets. The origin is \((0,0,0)\); a point on the \(x\)-axis is \((x,0,0)\), and a point in the \(YZ\) plane is \((0,y,z)\). The signs of the coordinates fix the octant. The distance between \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) is \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\); in particular the distance of \(Q\) from the origin is \(\sqrt{x_2^2+y_2^2+z_2^2}\). This formula is used to test collinearity, identify triangles and parallelograms, and find equations of point sets.
Coordinate axes and coordinate planes in spaceOctantsCoordinates of a point in spaceSign conventions and octant identificationDistance between two pointsApplications: collinearity and figures
Key terms
Coordinate axes in space
Three mutually perpendicular lines — the \(x\), \(y\) and \(z\)-axes — meeting at the origin.
Coordinate planes
The \(XY\), \(YZ\) and \(ZX\) planes determined by pairs of axes.
Octants
The eight parts into which the three coordinate planes divide space.
Coordinates of a point
An ordered triplet \((x,y,z)\) of perpendicular distances from the \(YZ\), \(ZX\), \(XY\) planes.
Origin
The point \((0,0,0)\) where the three axes intersect.
