Vector Equations

The angle between two planes

1

 

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

1

2 3

4

 

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

5

Example

Calculate the distance between the planes
          π1:       x + 2y - 2z = 5
and    π2:     6x + 12y - 12z = 8

      

6

7

8

9

Coplanar vectors

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

2

Vector equation of a plane

From the coplanar section above,
c=λa+μb

When position vectors are used,

22

 10

 

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)

11

If λ = 2 and μ =3

12

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+μ

33

The equations of a line

  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).

 

33

  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.

  13

14

15

  16 

17

18

19

 

Example

Find the vector equation of the straight line through
(3,2,1) which is parallel to the vector 2i +3j +4k

20

 

Example

Find the vector form of the equation of the
straight line which has parametric equations

21

 

Example

Find the Cartesian form of the line which has
 position vector  3i +2j +k  and is parallel to
the vector i - j + k

22

 

Example

  Find the vector equation of the line passing
  through A(1,2,3) and B(4,5,6)

23

24

   

 

Example

25

26

27

  

Example

28

  29

30

 

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

   31

Example

32

1)

33

34

36 37 38

35

2)

39

40

41

 

The intersection of two lines

Example

   42
   

43

44

45

46

47

48

49

 

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

 

50

51

52

53

 

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

      54

Example

Find the distance between the point ( 3,1,-2)
and the plane x +2y + 2z = - 4

55

56

57

Alternatively

58

 

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’

59

60

61

62

63

 

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

Example

70

123

 

A line of intersection

An infinite number of solutions exist

Example

55

66

68 67 44

Parametric equations

1221

Two lines of intersection

An infinite number of solutions

Example

76

71

Using the second row

72

Substitute into first row

77

Substitute into third equation

78

3

 

Three lines of intersection
Similar to above.
Examine each pair of planes in turn.

Example

82

23

 

321

A plane of intersection

Two redundant equations

Example

79

74

No consistency

1231

No  intersection

Example

80

81

No consistency

All planes are parallel

232

 

 

 

© Alexander Forrest