Prof. Dr. Faisel Ghazi Mohammed Mathematics (Calculus III) Lecturer: Prof. Dr. Faisel Ghazi Mohammed Email: faisel@scbaghdad.edu.iq faiselgm73@gmail.com 2017-2018 All rights reserved

Lecture 3 Planes CHECK IF THIS CAN BE IMPROVED.

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 11.2 564-566 (1-36) 2 The Dot Product 11.3 572-574 (1-77) 3 Lines and Planes in Space 11.6 592-593 (1-32) 4 Cylinders and spherical Coordinates 11.9 612-613 (1- 42) 5 Functions of Two or More Variables 12 Derivatives for functions of two or more variables 12.1 622-624 (1-46) 6 Limits 12.3 634-759 (1- 48) 7 Chain Rules 12.6 651-652 (1-34) 8 Double and Iterated Integrals over Rectangles 15 Multiple Integrals 13.1 679-680 (1-31) 9 Surface Area 13.6 704-705 (1-29)

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.

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

Normal vector orthogonal to every vector in plane

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

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

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

Example 2 Solution

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

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

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

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

END of Lecture

Thank you Any Questions ?

Appendix

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

d θ

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