Lines: Vector Equation
Only one line can pass through any two distinct points in 3-space. Assume that we have two points, and
, and there is a point
that lies in the same line. If
,
, and
, then a vector equation for the line is
(1)
There is an alternative vector equation for the line:
Parametric Equations
Consider the aforementioned vectors having the following components:
and we know that . So,
These equations are called parametric equations for the line passing through and
.
Symmetric Equation
Symmetric equations for the line passing through and
are defined as
(2)
Planes: Vector Equation
There is only one plane containing point
with a vector
normal to the plane. If
is any point on
, and
and
, then a vector equation of the plane is
(3)
Cartesian Equation
If a plane has the normal vector of and contains the point
, then the point-normal form of the equation of the plane is
(4)
8-Minute Lecture on Lines and Planes in 3D
Reference
Dennis G. Zill. Advanced Engineering Mathematics, edition. Jones
Bartlett Learning. 2016.