Finding the Vector Equation of a Line Using the Cross Product
A line can be defined in terms of the cross product using the fact that if
is the position vector of a particular fixed point on the line relative to the origin and
is the position vector of a general point on the line, then
is parallel to the line, or more precisely, the tangent vector of the line. This means that
(1) where
is the tangent vector along the line sine the cross product of parallel vectors is zero.
We can use this to find the vector equation of a line. Suppose we have the line given by vector equation
Take
The tangent vector
With
we have
To show this gives the same line as the vector equation for the line, we can put
as required by (1).
