Phy S Lecture 2 Goals of Lecture 2 Introduce Interactive Learning Segments and try a few The Language of Vectors: –Understand conventions used in denoting vectors. –Visualize vector operations both from the geometric and algebraic point of view. Dot Products: –Understand geometric and algebraic formulation of the vector dot product. –Understand some properties of the dot product

Phy S Lecture 2 What is a Vector A scalar quantity is one that is represented by a single number (e.g. Mass Length time temperature volume…) A vector is a quantity which has both magnitude and direction (e.g. displacement, velocity, force) Magnitude: How long is the vector Direction: angle counterclockwise from x-axis or other sensible description. Geometrically: we represent a vector as an arrow

Basics Equal Vectors: vectors are equal if they have the same magnitude and direction regardless of where the vector “starts” Opposite Vectors: Vector are opposite if the magnitude is the same but direction is opposite Unit Vectors: is in the direction of but of length 1.

Phy S Lecture 2 Vector Addition Geometrically: Parallel transport the tail of B to the head of A. The sum goes from the tail of A to the head of B. A+B=CA+B=C A (5,1) B(-2,4) C(3,5) Algebraically: Add the components A 5 1 B -2 4 C 3 5 Vector addition is commutative and associative

Phy S Lecture 2 interACTive learning segments (ACTs). Get your clickers ready I will post a question, you should talk to your neighbor and “vote” on the correct answer with your clicker You can change your answer if you change your mind. When you click, a box with your clicker number will appear At the end of the time, a bar graph will show the summary of results.

Phy S Lecture 2 ACT: Vector addition All the vectors below have the same magnitude. Which of the following arrangements will produce the largest resultant when the two vectors are added?

Phy S Lecture 2 The language of vectors To keep straight which variables are scalar quantities, it is conventional to draw a little arrow over vector variables. A B C If A is a vector, we use A (without arrow) to denote the magnitude. Sometimes boldface is used for vector and normal font for magnitude. Thus vector=A; magnitude=A More formally, we can indicate magnitude of a vector by vertical bars. Thus A=|A|.

Phy S Lecture 2 Vector Addition Geometric: A B C A+B=C Algebraic: A x +B x =C x A y +B y =C y A z +B z =C z Subtraction: A-B=D A B D -B Note two ways to think:  D is A plus –B  D goes from tip of B to tip of A when A and B are rooted at a common point

Phy S Lecture 2 Components The components of a vector can be thought of as the projections along the coordinate axes. These are sometimes called the Cartesian coordinates: We can denote A in terms of its components as: A=(A x,A y,A z ) x y A AxAx AyAy

Phy S Lecture 2 Unit Vectors A unit vector is a vector of length 1. To indicate a vector is a unit vector, we put a hat on it: Some special unit vectors –The unit vector that points along the x axis is denoted i –The unit vector that points along the y axis is denoted j –The unit vector that points along the z axis is denoted k ^ ^ ^ i j k ^ ^ ^ x y z Any vector can be written in terms of these basic unit vectors If A=(A x,A y,A z ) then A denotes a unit vector parallel to

Phy S Lecture 2 How do I remember which is sin and which is cos? Polar Notation In 2 dimensions, one can also describe a vector by its magnitude and direction. The direction is the angle  taken counterclockwise from the x axis  A |A|

Phy S Lecture 2 Dot Products Dot Products are the workhorse of vector analysis Definition Algebraic In terms of components: Geometric If  is the angle between A and B: A B  Where does this come from?

Phy S Lecture 2 Properties of The Dot Product The dot product takes two vectors as inputs and produces a scalar as output. The dot product is commutative: AB=BA Dot product distributes over vector addition: The dot product between a vector, A, and a unit, u, vector gives the projection of A along u. In particular, the components of A are the dot products with i, j and k The length of a vector can be expressed in terms of the dot product: |A|²=AA u A Au       

Phy S Lecture 2 Some Special Cases Vectors that are going in the same direction Vectors that are going in opposite directions Perpendicular vectors Remember vector #1 vector #2 Dot Product Scalar Note: Later in the course we will learn about another kind of product called the “cross product” which takes in two vectors and spits out another vector. Do not get them confused.

Phy S Lecture 2 Angle Between Vectors We can use the geometric definition of the dot product to determine the angle between two vectors: This tells the angle between A and B but not the direction of the angle A B 

Phy S Lecture 2 Example of Vector Algebra: Law of cosines Consider a triangle, The sum of the vectors representing the sides is 0 A BC C=-(A+B)C²=(A+B)²=(A+B)·(A+B) =A²+B²+2A·B =A²+B²+2AB cos(  ) =A²+B²-2AB cos(  )   Binomial expansion of dot product: a useful trick-learn it!