Math 1600 Lecture 3 1 of 5 Math 1600A Lecture 3, Section 002 Announceents: More texts, solutions anuals and packages coing soon. Read Section 1.3 for next class. Work through recoended hoework questions. Here are scans of the questions fro sections 1.1 and 1.2. Tutorials start Septeber 18, and include a quiz covering until Monday's lecture. More details on Monday. Bonus office hours: today, 2:30-3:30, MC103B. Also, if you can't ake it to y office hours, feel free to attend Hugo Bacard's office hours. Help Centers Monday-Friday 2:30-6:30 in MC 106 starting Sept 19. Questions after class: Questions about hoework probles should be asked at the Help Centers, tutorials or office hours. Questions about y lecture should be asked during class, as others are likely to have the sae question. (And it akes e feel like you are paying attention. :-) Questions after class should be liited to questions specific to you. There are too any students for e to answer general questions. Lecture notes (this page) available fro course web page by clicking on our class ties. Answers to lots of adinistrative questions are available on the course web page as well. Review of last lecture: Many properties that hold for real nubers also hold for vectors in 1.1. But we'll see differences later. R n : Theore v v 1, v 2,, v k Definition: A vector is a linear cobination of vectors if there
exist scalars,,, c 1 c 2 (called coefficients) such that We also call the coefficients coordinates when we are thinking of the vectors Vectors odulo : c k v = c 1 v 1 + + c k v k. v 1, v 2,, v k Math 1600 Lecture 3 2 of 5 as defining a new coordinate syste. Z = {0, 1,, 1} with addition and ultiplication taken odulo eans that the answer is the reainder after division by. Z 10 8 8 = 64 = 4 (od 10) For exaple, in,.. That Z n n Z is the set of vectors with coponents, each of which is in. New aterial To find solutions to an equation such as you can siply try all possible values of. In this case, and both work, and no other value works. 6x = 6 (od 8) x 1 5 Note that you can not in general divide in Z, only add, subtract and ultiply. Section 1.2: Length and Angle: The Dot Product Definition: The dot product of vectors and in is the real nuber defined by Since is a scalar, the dot product is soeties called the scalar product. The dot product will be used to define length, distance and angles in R n. Exaples on whiteboard, including in Z. Theore 1.2: For vectors in and in : (a) := u 1 v 1 + + u n v n. = v u u, v, w R n c R
(b) (c) (d) (e) u ( v + w ) = + u w (c u ) v = c( ) = u (c v ) u u 0 u u = 0 if and only if u = 0 Math 1600 Lecture 3 3 of 5 Again, very siilar to how ultiplication and addition of nubers works. Explain (b) and (d) on whiteboard. (a) and (c) are explained in text. Length fro dot product v v := v v = + +. Definition: The length or nor of is the scalar defined by Whiteboard: Pythagorean theore in R 2. Exaple in R 4. cv = c v Definition: A vector of length 1 is called a unit vector. Whiteboard: Unit vectors in R 2. Unit vector in sae direction as v, forula and exaple. in. Standard unit vectors in. Picture for triangle inequality. Theore 1.5: The Triangle Inequality: For all and in, Whiteboard: Exaple in : and. Distance fro length Thinking of vectors and as starting fro the origin, we define the distance between the by the forula v 2 1 v generalizing the forula for the distance between points in the plane. Exaple: The distance between and is v 2 n e 1, e 2, e 3 R 3 R n u + v u + v u R 2 [1, 0] [3, 4] v d( u, v ) = u v = ( u 1 v 1 ) 2 + + ( u n v n ) 2, u = [10, 10, 10, 10] v = [11, 11, 11, 11]
Angles fro dot product The unit vector in at angle fro the -axis is. Notice that More generally, given vectors and in, one can show using the law of cosines that where θ (1 ) 2 + (1) 2 + (1) 2 + (1) 2 = 4 = 2. R 2 θ x u = [cos θ, sin θ] = [cos θ, sin θ] [1, 0] = 1 cos θ + 0 sin θ = cos θ. u e 1 u v R 2 = cos θ u v, is the angle between the. u v cos θ 1 In particular,, since. Math 1600 Lecture 3 4 of 5 This holds in R n as well: Theore 1.4: The Cauchy-Schwarz Inequality: For all and in, We can therefore use the dot product to define the angle between two vectors and v in R n by the forula where we choose -1 and 1. u v cos θ =, i.e., θ = arccos( ), u v u v 0 θ 180 Section 1.2 continued in lecture 4.. This akes sense because the RHS is between u
Math 1600 Lecture 3 5 of 5