be the position vector of a point P in space. Then the
vector equation of a line passing through point P in a direction given by vector
is
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LINES AND PLANES IN SPACE
Equation of a line in space. Let
be the position vector of a point P in space. Then the
vector equation of a line passing through point P in a direction given by vector
is
where t is a parameter that ranges over the real numbers. If
then 1) is equivalent to the following parametric representation of a straight line:
As the parameter t varies over the real numbers the vector
generates the line.
Equation of a plane in space. Let
be the position vector of a point P in space. Then the
vector equation of a plane passing through point P and parallel to two noncollinear vectors
and
is
where parameters u and v range independently over the real numbers. If
then 3) is equivalent to the following parametric representation of a plane:
As the parameters u and v vary over the real numbers the vector
generates the plane.
Normal and one point form of the equation of a plane. Let
be the position vector
of a point P in space. Then the vector equation of a plane passing through point P and
perpendicular to some vector
(i.e. a normal to the plane) is
If
5) is equivalent to
6) (x1 - a1)n1 + (x2 - a2)n2 + (x3 - a3)n3 = 0
Three point form of the equation of a plane. Let
be the position vectors of
three noncollinear points P1, P2 and P3 in space. Then the vector equation of the plane passing
through points P1, P2 and P3 is
Proof. The vectors
and
represent two linearly independent vectors in the plane and
so
represents a vector normal to the plane. Equation 7) then follows from
equation 5) above.
If
7) is equivalent to
Reference.
Lipschutz. Differential Geometry. Chap. 2
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