Equation
of a plane passing through three given non-collinear points
If three non-collinear points with position vectors ,
,
are
given, then the vector equation of the plane passing through the given points
in parametric form is
Consider a plane passing through three non-collinear points A,
B, C with position vectors ,
,
respectively.
Then atleast two of them are non-zero vectors. Let us take
≠0 and
≠0 . Let
be the position vector of an arbitrary point P on the
plane. Take a point D on AB (produced) such that
is
parallel to
and
is parallel to
. Therefore,
This is the parametric form of vector equation of the plane
passing through the given three non-collinear points.
Let A, B, and C be the three non collinear
points on the plane with position vectors ,
,
respectively. Then atleast two of them are
non-zero vectors. Let us take
≠0 and
≠0 .
Now =
-
and
=
-
. The
vectors (
-
and
-
lie on the
plane. Since
,
,
are non-collinear,
is not parallel to
.
Therefore,
is perpendicular
to the plane.
If is the position vector of an arbitrary point
P(x, y, z)
on the plane, then the equation of the plane passing through the point A with
position vector
and perpendicular to the vector
is given
by
This is the non-parametric form of vector equation of the plane
passing through three non-collinear points.
If (x1 , y1 , z1
), (x2 , y2 , z2 ) and (x3
, y3 , z3 ) are the coordinates of three
non-collinear points A, B, C with position vectors ,
,
respectively and (x, y, z) is
the coordinates of the point P with position vector
, then we have
Using these vectors, the non-parametric form of vector equation
of the plane passing through the given three non-collinear points can be
equivalently written as
which is the Cartesian equation of the plane passing through
three non-collinear points.
Related Topics
Privacy Policy, Terms and Conditions, DMCA Policy and Compliant
Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.