## Quick Reference

(in three dimensions)

As in two dimensions there are a number of standard formats which can be used. With the exception of the two-point form the 2-dimensional forms are not extensible to three dimensions, while the parametric and vector forms given below can be reduced to two dimensions, but are not commonly used. The following refer to the equation of the line through *A* (*x*_{1}, *y*_{1}, *z*_{1}) and *B* (*x*_{2}, *y*_{2}, *z*_{2}).

Two-point form.

The condition that the general point *P* (*x*, *y*, *z*) lies on the line through *AB* is equivalent to requiring the direction ratios of *AP* = the direction ratios of *AB*.

Parametric form.

If the vector is parallel to *AB* then the point *P* can be expressed in the form (*x*_{1}+*λ**l*, *y*_{1}+*λ**m*, *z*_{1}+*λ**n*), in terms of the parameter *λ*.

Vector form.

If the vector is parallel to *AB* then the position vector of the point *P* can be expressed in the form from which the parametric form can be derived immediately. If is the position vector of the point *A*, and **b** is defined similarly, the vector form can be written as **r**=**a**+*λ*(**b**−**a**) or alternatively defined by (**r**−**a**)×(**b**−**a**)=0. This last form is relying on the fact that *A*, *B*, *P* are in a straight line, so the angle between *AB* and *AP* is zero, and therefore the vector product will be zero.

*Subjects:*
Mathematics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.