Calculus and Vectors – How to get an A+
9.4 Intersection of three Planes
A Intersection of three Planes
Let consider three planes given by their Cartesian
equations:
π 1 : A1 x + B1 y + C1 z + D1 = 0
π 2 : A2 x + B2 y + C 2 z + D2 = 0
π 3 : A3 x + B3 y + C3 z + D3 = 0
B Unique Solution
(Point Intersection – Non Coplanar Normal Vectors)
In this case:
The point(s) of intersection of these planes is (are) related
to the solution(s) of the following system of equations:
⎧ A1 x + B1 y + C1 z + D1 = 0
⎪
⎨ A2 x + B2 y + C 2 z + D2 = 0 (*)
⎪A x + B y + C z + D = 0
3
3
3
⎩ 3
There are three equations and three unknowns.
You may use substitution or elimination to solve this
system.
Ex 1. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
(1)
⎧ x − 3 y − 2 z = −9
⎪
(2)
⎨2 x − 5 y + z = 3
⎪− 3x + 6 y + 2 z = 8 (3)
⎩
Ö The planes intersect into a single point.
Ö The normal vectors are not coplanar:
r r r
n1 ⋅ (n2 × n3 ) ≠ 0 .
Ö By solving the system (*), you get a unique
solution for x , y , and z .
C Infinite Number of Solutions (I)
(Line Intersection – Non Parallel Planes and
Coplanar Normal Vectors)
Ex 2. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
⎧ x + y + 2 z = −2
⎪
⎨3x − y + 14 z = 6
⎪ x + 2 y = −5
⎩
9.4 Intersection of three Planes
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Calculus and Vectors – How to get an A+
In this case:
Ö The planes are not parallel but their normal
r r r
vectors are coplanar: n1 ⋅ (n2 × n3 ) = 0 .
Ö The intersection is a line.
Ö One scalar equation is a combination of the other
two equations.
Ö By solving the system (*), you may express two
variables in terms of the third one using two
equations.
D Infinite Number of Solutions (II)
(Line Intersection – Two Coincident Planes and one
Intersecting Plane)
Ex 3. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
(1)
⎧x + y − z = 2
⎪
(2)
⎨x − 2 y + z = 4
⎪2 x − 4 y + 2 z = 8 (3)
⎩
In this case:
Ö Two planes are coincident and the third plane is
not parallel to the coincident planes.
Ö The intersection is a line.
Ö Two scalar equations are equivalent. The
coefficients A, B, C , D are proportional for these
two equations.
Ö You may express two variables in terms of the
third one using two non equivalent equations.
E Infinite Number of Solutions (III)
(Plane Intersection – Three Coincident Planes)
In this case:
Ex 4. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
⎧x − y − 2z = 1
⎪
⎨2 x − 2 y − 4 z = 2
⎪− 4 x + 4 y + 8 z = −4
⎩
Ö The coefficients A, B, C , D are proportional for all
three equations.
Ö Any point of one plane is also a point on the other
two planes.
Ö The intersection is a plane.
9.4 Intersection of three Planes
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Calculus and Vectors – How to get an A+
Ex 5. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
(1)
⎧ x + 2 y + 3z = 1
⎪
2
x
4
y
6
z
1
(
2)
+
+
=
−
⎨
⎪− x − 2 y − 3z = 3 (3)
⎩
F No Solution
(Parallel and Distinct Planes)
In this case:
Ö There are three parallel and distinct planes.
Ö There is no point of intersection.
Ö There is no solution for the system of equations
(the system of equations is incompatible).
Ö The coefficients A, B, C are proportional but the
coefficients A, B, C , D are not proportional.
Ö By solving the system (*) you get false
statements (like 0 = 1 ).
Ex 6. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
(1)
⎧x + y − z = 1
⎪
(2)
⎨x + y + z = 2
⎪− 2 x − 2 y + 2 z = 3 (3)
⎩
G No Solution (II)
(H Configuration)
In this case:
Ö Two planes are parallel and distinct and the third
plane is intersecting.
Ö There is no point of intersection.
Ö The coefficients A, B, C are proportional for two
planes.
Ö There is no solution for the system of equations
(the system of equations is incompatible).
Ö By solving the system (*) you get false
statements (like 0 = 1 ).
Ex 7. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
(1)
⎧ x + 2 y + 3z = 1
⎪
+
+
=
3
x
6
y
9
z
3
(
2)
⎨
⎪− 2 x − 4 y − 6 z = 2 (3)
⎩
H No Solution (III)
In this case:
Ö Three planes are parallel but only two are
9.4 Intersection of three Planes
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Calculus and Vectors – How to get an A+
coincident.
Ö The coefficients A, B, C are proportional for all
equations but the coefficients A, B, C , D are
proportional only for two planes.
Ö There is no solution for the system of equations
(the system of equations is incompatible).
Ö By solving the system (*) you get false
statements (like 0 = 1 ).
Ex 8. Solve the following system of equations. Give a
geometric interpretation of the solution(s).
(1)
⎧2 x + y + z = 1
⎪
⎨− x + y + z = −1 (2)
⎪x + y + z = 0
(3)
⎩
I No Solution (IV)
(Delta Configuration)
In this case:
Ö The planes are not parallel (the coefficients
A, B, C are not proportional).
Ö The normal vectors are coplanar
r r r
( n1 ⋅ (n2 × n3 ) = 0 ).
Ö There is no point of intersection between all three
planes.
Ö There is no solution for the system of equations
(the system of equations is incompatible).
Ö By solving the system (*) you get false
statements (like 0 = 1 ).
Reading: Nelson Textbook, Pages 520-529
Homework: Nelson Textbook: Page 530 #8, 9, 10, 13
9.4 Intersection of three Planes
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