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EXAMPLE 6 Line of Intersection of Two Planes
Find parametric equations for the line of intersection of the planes
Solution that satisfy the two equations in the system
The line of intersection consists of all points
Solving this system by Gaussian elimination gives , , . Therefore, the line of intersection can be
represented by the parametric equations
Vector Form of Equation of a Line
Vector notation provides a useful alternative way of writing the parametric equations of a line: Referring to Figure 3.5.5, let
be the vector from the origin to the point , let be the vector from the origin to the point
, and let be a vector parallel to the line. Then , so Formula 6 can be rewritten as
Taking into account the range of t-values, this can be rewritten as (8)
This is called the vector form of the equation of a line in 3-space.
Figure 3.5.5
Vector interpretation of a line in 3-space.
EXAMPLE 7 A Line Parallel to a Given Vector
The equation
is the vector equation of the line through the point that is parallel to the vector .
