THE SCALAR EQUATION OF A PLANE
•
In general can the equation of a plane be given
as:
³
3 +
´
4 +
²
5 = 8
And the vector (
a
,
b
,
c
) is perpendicular to the
plane

9/2/2015
4
THE SCALAR EQUATION OF A PLANE
We can then write the scalar equation of the plane as:
µ − µ
:
. 0 = 0
With:
µ = +,-./.,0 12²/,µ =
3, 4, 5
µ
:
= ³ +,.0/ /6³/ 7.2- ,0 /62 +7³02 e. g. =
x
:
, y
:
, z
:
0 = /62 12²/,µ +2µ+208.²97³µ /, /62 +7³02 =
(0
@
, 0
A
, 0
B
)
Thus:
µ − µ
:
. 0 = 0
3, 4, 5
−
3
:
, 4
:
, 5
:
.
0
@
, 0
A
, 0
B
= 0
0
@
3 − 3
:
+ 0
A
4 − 4
:
+ 0
B
5 − 5
:
= 0
THE SCALAR EQUATION OF A PLANE
µ − µ
:
. 0 = 0
3, 4, 5
−
3
:
, 4
:
, 5
:
.
0
@
, 0
A
, 0
B
= 0
0
@
3 − 3
:
+ 0
A
4 − 4
:
+ 0
B
5 − 5
:
= 0
0
@
3 − 0
@
3
:
+ 0
A
4 − 0
A
4
:
+ 0
B
5 − 0
B
5
:
= 0
0
@
3 + 0
A
4 + 0
B
5 = 0
@
3
:
+ 0
A
4
:
+ 0
B
5
:
In the same format as:
³3 + ´3 + ²4 = 8

9/2/2015
5
EXAMPLES
Example 2.
The two vectors
³
=
1
2
2
and
b
=
1
−2
1
are
both in a certain plane. The point
(1,1,0)
is also in the
plane.
(a)
Find the vector equation of the plane
.
(b) Find the parametric equations of the plane.
(c)
Find the equation of the plane in terms of
C, D
and
E
only.
Solution:
(a):
µ
=
3
4
5
=
1
1
0
+ ¸
1
2
2
+ ¹
1
−2
1
.
(b):
3 = 1 + ¸ + ¹
4 = 1 + 2¸ − 2¹
5 = 2¸ + ¹
(c) Find the equation of the plane in terms of
C, D
and
E
only.
Solution:
Parametric equations are
3 = 1 + ¸ + ¹
(1)
4 = 1 + 2¸ − 2¹
(2)
5 = 2¸ + ¹
(3)
(1):
¸ + ¹ = 3 − 1
(3):
2¸ + ¹ = 5
Subtract:
¸ = 5 − 3 + 1.

#### You've reached the end of your free preview.

Want to read all 9 pages?

- Summer '20
- Vectors, Vector Space, Force, Euclidean vector