Straight line through (with position vector ) and parallel to a vector . Let be a general point on , then . Since is parallel to , hence (for some scalar ), may be positive or negative. Thus . This is the vector equation of a straight line.
If , and , the equation
gives Equality of the vectors gives 3 scalar equations, or , or and or . Since , (for different points on L), we find that these three scalar equations give the Cartesian equations of L as
This is called the standard or canonical form.
Example 3.14:
Find the position vector of a point on a
straight line L and a vector along L whose Cartesian
equations are
.
the standard form of L is
Point : , position vector of A: . (parallel to L)
Example 3.15:
Example: Find the Cartesian equations of
a straight line L through the point
in the direction of the vector
.
L:
gives
. This gives the following Cartesian equations of
L: