The angle between two planes is found
using the scalar product.
It is equal to the acute angle determined by
the normal vectors of the planes.
Example
Calculate the angle between the planes
π1: x +2y -2z = 5
and π2: 6x -3y +2z = 8
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
If a relationship exists between the vectors a, b and c
such that c=λa+μb, where λ and μ are constants,
then vectors a, b and c are co-planar.
If three vectors are co-planar,
c=λa+μb
From the coplanar section above,
c=λa+μb
When position vectors are used,
r=(1-λ-u)a+ λb+μc is the vector equation of the plane.
Since λ and b are variable, there will be many possible equations for 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)
If λ = 2 and μ =3
When A is a known point on the plane,
R is any old point on the plane and b and c are vectors
parallel to the plane,
the vector equation of the plane is
r=a+λb+μc
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).
uis 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.
Example
Find the vector equation of the straight line through
(3,2,1) which is parallel to the vector 2i +3j +4k
Example
Find the vector form of the equation of the
straight line which has parametric equations
Example
Find the Cartesian form of the line which has
position vector 3i +2j +k and is parallel to
the vector i - j + k
Example
Find the vector equation of the line passing
through A(1,2,3) and B(4,5,6)
Example
Example
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
1)
2)
Example
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
To find the distance of a point P to a plane
Alternatively
Example
Find the distance between the point ( 3,1,-2)
and the plane x +2y + 2z = - 4
Alternatively
To find the distance of a point P to a Line L
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
Example
A line of intersection
An infinite number of solutions exist
Example
Parametric equations
Two lines of intersection
An infinite number of solutions
Example
Using the second row
Substitute into first row
Substitute into third equation
Three lines of intersection
Similar to above.
Examine each pair of planes in turn.
Example
A plane of intersection
Two redundant equations
Example
No consistency
No intersection
Example
No consistency
All planes are parallel