Day 1: Introduction to Vectors + Vector Arithmetic A is a quantity that has magnitude but no direction. You can have signed scalar quantities as well. A is a quantity that has both magnitude and direction. Some examples of scalar quantities are: Some examples of vector quantities are: To specify the velocity of a moving point in the coordinate plane, we must give both the rate at which the point moves (speed) and the direction of that motion. The velocity vector incorporates both pieces of information. The length of the vector represents its magnitude and its direction is represented by the direction of the arrow that represents the vector. A vector u is represented in boldface, but for our purposes, we will use u. There are a number of different ways to express a vector. When we describe vectors, it is usual to do so by expressing them in terms of a. To describe vectors in two dimensions, we use two basis vectors, i, and j. j i To describe a vector that is 2 steps to the right and 3 steps up we write To describe a vector that is 1 step to the left and 4 steps up we write The vectors column vector i and 4 j are known as components of the vector b. It can also be written as a In column representation (in two dimensions), 1 i 0 and j. 0 1 i, j, <a,b>, and (a,b). These are for two- Other possible representations for vectors are dimensional vectors.
To describe vectors in three dimensions, we use three basis vectors, i, j, and k. j k i Note that the component vectors are all at right angles to each other. (Imagine k is coming out of the page towards you). To represent vectors in higher dimensional space, simply add more basis vectors. In three dimensions, the notation for the vector u is u, u, In column representation, 1 i 0 0, 0 j 1 0, and 0 k 0 1 a b c., ai bj ck, <a,b,c>, or (a,b,c). Drawing Vectors in 2D Consider the vector from the origin O to the point A. We call this the vector of point A. O a A This position vector can be represented by a or a OA. The magnitude or length of can be represented by OA or a or a. or The vector is a position vector of B relative to A. It is a displacement. B A Ex. Draw the vector 3 b 3i 4 j. Note: In column form, this is. 4
Ex. Draw the vector a 2i 5 j. Note: Vectors do not have to be drawn starting from the origin. If they are, it is a special case that we will talk about later. Two vectors are equal only if they have the same magnitude (length) and direction. Therefore, equal vectors must be parallel. A negative vector has the same magnitude but opposite direction. For example, the same length but have opposite directions. Therefore, is the negative of BA and. BA have Note: Two vectors can have the same magnitude but different directions. Two vectors can also have the same direction (parallel) but different magnitude, or two vectors can have the same magnitude, be parallel, but be in opposite directions. (Show each case). Ex. Given the coordinates of point A and the components of 3 A and 7 4 2, find point B.
Vector Arithmetic Adding and Subtracting Sum and Difference We can add vectors graphically or algebraically. To add vectors graphically, we add them nose to tail. If a 2i 3 j and b i 4 j represent a b in the plane. The zero vector is 0 0 and for any vector a, a + (-a) = (-a) + a = 0. 0 As mentioned before, the negative of a vector is the same magnitude but in the opposite direction. Subtraction of a vector therefore is the same as adding the opposite.
Ex a) Represent a b in the plane if a 2i j and b i 3 j. (This is a ( b) ) Is b a the same as a b? Check by graphing. Vectors can be added algebraically by adding their respective components. b) Add a + b algebraically. Check to see if this looks like the graph. c) Find a b. Check the graph. Ex. If 4 a 1 and b, find a b. 5 2
Vectors in three dimensions are added and subtracted the same way that vectors in two dimensions are. Ex. If 3 p 1 4 and 2 q 0 3, find p q and q p. Vector Arithmetic Scalar Multiplication In addition to being added and subtracted, vectors can be multiplied by a scalar (non-vector quantity). Multiplication by a scalar affects the length or magnitude of a vector, but not the direction (unless the scalar is negative). ka 1 If k is a scalar, then ka =. Each component is multiplied by the scalar. ka 2 Ex. If 1 a 2 and b 3, find: 5 a) 4b b) a 2b 1 c) b 2a 2