2. Prof. Dr. Faisel Ghazi Mohammed
Mathematics
(Calculus III)
Lecturer:
Prof. Dr. Faisel Ghazi Mohammed
All rights reserved

3. Lecture 3
Planes
CHECK IF THIS CAN BE IMPROVED.

4. 11 12 15 Book: Material Chapter
“Calculus, with Differential Equations”, by Varberg etl ,9th ed, 2006.
Material
Chapter
Section
Pages
Exercises 1
Vectors 11
Geometry in Space and Vectors
11.2
(1-36) 2
The Dot Product
11.3
(1-77) 3
Lines and Planes in Space
11.6
(1-32) 4
Cylinders and spherical Coordinates
11.9
(1- 42) 5
Functions of Two or More Variables 12
Derivatives for functions of two or more variables
12.1
(1-46) 6
Limits
12.3
(1- 48) 7
Chain Rules
12.6
(1-34) 8
Double and Iterated Integrals over Rectangles 15
Multiple Integrals
13.1
(1-31) 9
Surface Area
13.6
(1-29)

5. Learning Objective
Specify different sets of data required to specify a line or a plane.
Memorize formulae for parametric equation of a line in space and explain geometrical and physical interpretations.
Memorize formulae for parametric, level set, and graph-of-function descriptions of plane in space and provide geometrical interpretations with the aid of sketches. Convert between these algebraic representations of a plane and recognize which to apply in problem solving contexts.
Sketch specific lines and planes described using algebraic formulae
Solve problems involving geometric relationships between lines and/or planes.

6. Define three-dimensional planes Using vectors.
In this section, we will learn how to:
Define three-dimensional planes
Using vectors.

7. Normal vector orthogonal to every vector in plane

8. Arithmetic definition of Dot Product
The standard form for the equation of a plane

9. Example 1
Find the equation of a plane that goes through the origin with normal vector
Solution
A, B, C
eq. of plane
Note: if I just gave you this plane eq., we immediately know <1,2,3> is normal vector

10. Example 2
Find the equation of the plane through (1,-3,4) perpendicular to
x1,y1,z1
A, B, C
Note:
For any given plane, the most important feature of the normal vector is the direction.
Therefore, we can use any scaled version of the normal vector when determining
i.e., normal vector is not unique
Solution

11. Example 2
Solution

12. See Appendix for more information
(P0 not on plane)
See Appendix for more information

13. Example 3 -3x +2y + z = 9 and 6x - 4y - 2z = 19. Solution
Find the distance between the parallel planes
-3x +2y + z = 9 and 6x - 4y - 2z = 19.
(1)
(2)
A B C D
Solution
Find a pt on plane (1) , then find the distance from that pt to plane (2)
On plane (1), P0(0, 0, 9)
x0 y0 z0
A B C

14. Example 4 Solution Find the (smaller) angle between the two planes
-3x +2y + 5z = 7 and x - 2y - 3z = 2.
(1)
(2)
Solution
Note: this is equivalent to finding angle between the 2 normal vectors
Side View

15. Example 4 Solution θ=210 smaller angle between planes
1590
θ=210 smaller angle between planes
Remember that the angle between normal vector is exactly the angle between 2 planes

16. END
of Lecture

17. Thank you
Any Questions ?

18. Appendix

19. (abs. value to make sure its positive length)
(P0 not on plane)
From trigonometry
(abs. value to make sure its positive length)

20. d θ

21. d θ
But P1 is on the plane :
 Ax1+By1+Cz1=D