Vector Equations
· The angle between two planes
The angle between two planes is found
using the scalar product.
Example
Calculate the angle between the planes
π1: x +2y -2z = 5
and π2: 6x -3y +2z = 8
· The distance between parallel planes
Let P be a point on plane π1 : ax +by +cz = n
a.x = n
and Q be a point on plane π2 : ax +by +cz = m
a.x = m
Since the planes are parallel, they share the common normal, a
a=(ai +bj+ck)
The distance between the planes is
Example
Calculate the distance between the planes
π1: x +2y -2z = 5
and π2: 6x +12y -12z = 8
· Coplanar vectors
If a relationship exists between the vectors a, b and c
such that c=λa+μb
the a, b and c are co-planar.
If three vectors are co-planar,
c=λa+μb
· vector equation of a plane
If a, b and c are position vectors on a plane
r=a+λb+μc is the vector equation of the plane.
A is a point on the plane, b and c are vectors
parallel to the plane.
When the position vectors are used,
r=(1-λ-u)a+ λb+μc is the vector equation of the plane.
Example
Find a vector equation of the plane through the points
A(-1,-2,-3) , B(-2,0,1) and C(-4,-1,-1)
A line can be described when a point on it
and
its direction vector a vector parallel to the line
are known.
In the diagram below, the line L passes
through points A(x1,y1,z1)
and P (x,y,z).
u is the
direction vector ai +bj +ck
Being on the line, it has the same direction as
any
parallel line.
O is the origin.
a and p
represent the position vectors of A and P.
Examples
Find the vector
equation of the straight line through
(3,2,1) which
is parallel to the vector 2i +3j +4k
Find the vector
form of the equation of the
straight line
which has parametric equations
Find the Cartesian
form of the line which has
position vector 3i
+2j +k and is parallel to
the vector i - j + k
Find the vector equation of the line passing
through A(1,2,3) and B(4,5,6)
·
The angle between a line and a plane
The angle θ between a line
and a plane is the
complement of the angle
between the line and
the normal to the plane.
If the line has direction
vector u and the
normal to the plane is a, then
Example
·
the intersection of two lines
Example
·
The intersection of two planes
To find the
equations of the line of intersection
of two planes, a direction vector and point
on the line is
required.
Since the line
of intersection lies in both planes,
the direction
vector is parallel to the vector products
of the normal of
each plane.
Example
Find
the equation for the line of intersection
of
the planes
-3x
+2y +z = -5
7x
+3y -2z = -2
·
The distance from
a point to a plane
To find the distance of a
point P to a plane
1) Find the equation of
the projection PP’ by using
the normal to the plane
and the point P.
2) Find the co-ordinates
of P’ ,the intersection
with the plane.
3) Apply the distance
formula to PP’
Alternatively
Example
Find the distance between
the point ( 3,1,-2)
and the plane x +2y +2z =
-4
Alternatively
·
The distance from a point to a line
To find the distance of a
point P to a Line L
1) Let the line have
direction vector u and parameter λ
2)
Find the co-ordinates of PP’ by using
the scalar product with u
and the point P.
3) Apply the distance
formula to PP’
·
The intersection of three planes
To solve the intersection,
use the equations of the
plane ax +by +cz +d = 0
to form an augmented
matrix, which is solved
for x, y and z.
The intersection between
three planes could be
A single point
A unique solution is found
A line of intersection
An infinite number of
solutions
Two lines of intersection
An infinite number of
solutions
Three lines of intersection
Similar to above.
Examine each pair of
planes in turn.
A plane of intersection
Two redundant equations
No intersection