MAT 188-WEEK 1 Sets, Vectors, Lines, Planes (c)2022 C. Karimianpour Learning Objectives: • • • • Familiarity with set notation Geometric understanding of vectors in two and three dimensions Fluency in performing vector algebra and interpreting it geometrically 2 3 Geometric and Algebraic understanding of lines and planes in R and R . Vocabulary: set, vector, vector addition and scalar multiplication, dot product, orthogonal vectors, parallel vectors, line, plane, vector form, parametric form. direction vector, normal vector. Reading from the textbook: : Appendix A up to cross product. Specically make sure to ponder upon Denition A.1, Theorem A.2, Denition A.3, Denition A.4, Theorem A.5, Denition A.6, Denition A.8. Sec 1.1 1 MAT 188-WEEK 1 Sets, Vectors, Lines, Planes 2 Sets A set is a collection of objects. The objects in a set are called elements or members of that set. For instance, the collection of all Toronto based sports teams is a set and the Blue Jays is a member of this set, or the collection of all even numbers is a set and 2022 is a member of this set. Let S be a set. We a ∈ S to signify that the object a is an element of the set S and write b ∈ / S to signify that the b is not an element of the set S . For example, suppose E denotes the set of all even numbers. Then 4 ∈ E and 3 ∈ / E . Since a set has no distinguishing feature other than its content, there is a unique set containing no elements which is called the empty set and is denoted by ∅ or {}. Some other common write object 1 sets have specic names and notations: R: Z: Q: N: C: the set of all real numbers the set of all integers or whole numbers the set of all rational numbers the set of all natural numbers the set fo all complex numbers In this course, we refer to elements of R scalars. by There are multiple ways of describing a set. One way is to describe the set in words by specifying its members in words, like what we did for examples above. If the set is nite, and has manageable number of elements, we can describe it by listing all its members, separated by a comma, inside a set of curly brackets. For instance the set M containing the names of the following four mathematicians: Gauss, Noether, Mirzakhani and Viazovska can described as M = {Noether, Gauss, Mirzakhani, Viazovska}. Note that the order in which the elements are listed does not matter. Often time, listing all the elements is not an ecient, or even feasible, way of describing a set. For instance think about the set of all even integers we described above. It is impossible to list them all. A better way to describe sets is using the set builder notation. Let set P S be a set. We can dene another using the following general format P = {s ∈ S | s satises a certain property} To see this methods in action, let's describe the set E E = {z ∈ Z | z = 2k , Which reads E is the set of all is the set of all z in Z such that z z in Z such that z of even numbers for some integer k} is some integer multiple of two. In other words, E is even!). Another common way of describing a set is to give a parametric description of a typical element in that set: P = {an expression describing a typical element in Let's describe E P| specifying the parameters used in the description} again, using the second method E = {2k | k ∈ Z}, Which reads E is the set of all multiples of 2k 's where k is an integer. In other word, E is the set of all integer 2. Denition (Subset). We say a set A is a subset of a set B , and write A ⊆ B , if all the elements of A are also in B . In other words, A ⊆ B , if for every a ∈ A, a ∈ B . Denition (Equality of Sets). We say sets A and B are equal if A is a subset of B and B is a subset of A. That is A = B if A ⊆ B and B ⊆ A. 1You are already familiar with some of these sets, like R and Z, you maybe less familiar with some others like Q and C. Don't worry! MAT 188-WEEK 1 Denition (Union and Intersection of Sets) Sets, Vectors, Lines, Planes 3 . Let X be a set and A and B be subsets of X . The union of A and B is a set that contains all elements of A and B , that is A ∪ B = {x ∈ X | x ∈ A or x ∈ B} The intersection of A and B is the set of all common elements between A and B that is A ∩ B = {x ∈ X | x ∈ A and x ∈ B}. Example. Let A = {2, 5, 7, π} and B = {4, π, 5} be subsets of R. Then A ∪ B = {2, 5, 7, π, 4} and A ∩ B = {5, π}. MAT 188-WEEK 1 Sets, Vectors, Lines, Planes 4 Vectors: Algebraic View A matrix is a rectangular array of entries. 1 2 π −1 0 4 M is a 1 2 2 × 3 matrix . A column vector is a matrix with only one column and multiple rows for instance 2 is a 3 column vector. A row vector is a matrix with one row and multiple columns for instance 1 2 3 π is numbers. For instance M = For the purposes of our course, these entries are often is a matrix with two rows and three columns. We say a row vector. In this course, unless otherwise stated, by a vector we mean a column vector. The entries components. The collection of all column vectors with n components is denoted n-dimensional Euclidean vector space. That is of a vector are called its n n by R ; we will refer to R as the a1 a2 Rn = { ... | a1 , · · · , an are in R} an Fix n. Rn . Given ⃗v , w ⃗ ∈ Rn v1 w1 v1 + w1 v2 w2 v2 + w2 ⃗v + w ⃗ = . ... + ... = . . We can add, and scale vectors in vn Given vector ⃗v ∈ R n and a scalar wn vn + wn k∈R v1 kv1 v2 kv2 k⃗v = k ... = ... vn Reading from the textbook: kvn You should read Denition A.1, and Theorem A.2 and the following Examples in Appendix A to see the rules that govern vector addition and scalar multiplication. 2This is a naive denition for a matrix that we go with for now; soon in this course we will learn that matrices are WAY more than just an array of numbers. MAT 188-WEEK 1 Sets, Vectors, Lines, Planes 5 Vectors: Geometric View While it is dicult (or impossible) to visualize vectors with more that three components, We can 2 3 2 visualize vectors in R, R and R . Let's walk through visualizing vectors in R . Given two points P = (p1 , p2 ) xy -plane), consider an arrow, that is a directed line segment, denoted by starting from P and ending at Q. This arrow represents the displacement from P to Q. For instance, if P = (1, 1) and Q = (3, 2), P⃗Q can be understood as describing a walk we take in the plane where we move two units horizontally and one unit 2 2 vertically. The vector P⃗Q encodes this walk, and is denoted by P⃗Q = . Note that is just a 1 1 and Q = (q1 , q2 ) on the Cartesian plane (you might know this as P⃗Q direction that says move two units horizontally and one unit vertically. We can allow ourselves to begin walking from any point in the plane, and our end points will depend on where we start. In particular if we start from P we end up on the vector 2 1 Q. Where do we end up if we start from the origin and take a ride with ? We will most frequently begin at the origin. The x standard representation of a vector ⃗x = 1 x2 in the Cartesian coordinate plane is an arrow (a directed line segment) connecting the origin to the point (x1 , x2 ). We say A vector in ⃗x R2 is the position vector of the point x = (x1 , x2 ). (in standard representation) is uniquely determined by its endpoint. Conversely, with each point in the plane we can associate its position vector, which connects the origin to the given point. 2 In this sense, we can identify R , the set of all vectors with two components, with the set of all points on 2 the Cartesian plane. That is, it is safe to think of R as the Cartesian plane! With this set up, we can think of adding two vectors as walking along two vectors, one after the other. Visualizing vector addition this ways is often referred to as the parallelogram Law. Reading from the textbook: Read Geometrical representation of vectors in appendix A. Exercise. Pick two vectors ⃗v and w⃗ in R3 . Draw a picture that represents ⃗v and w⃗ geometrically. You can use Figure 7 and Figure 8 in Appendix A as a model. Draw ⃗v + w ⃗ is two ways, one using the parallelogram law and once by drawing w ⃗ from the tip of ⃗v . Explain why the two methods give us the same result. MAT 188-WEEK 1 Sets, Vectors, Lines, Planes 6 More on Vectors Once we think about vectors geometrically as arrows, we can talk about geometric concepts such as the length of vectors, parallel vectors and perpendicular vectors understood through our spacial conception n of these terms. Interestingly, we can make sense of these concepts for vectors in R , for any n, despite 3 the fact that we can not draw or visualize vectors above R . x1 x2 n The length or norm ∥⃗x∥ of a vector ⃗x = ... in R is Denition (Norm of a Vector). xn ∥⃗x∥ = Denition . We say that two vectors ⃗v and w⃗ in Rn are parallel if one of them is a (Parallel Vectors) scalar multiple of the other. Reading from the textbook: Exercise. (1) (2) (3) q x21 + x22 + · · · + x2n Read Example 3 and Example 4 in Appendix A. Draw a vector ⃗v in Rn . Visually justify how the denition of the norm of a vector matches your intuition of the length of the vector you drew. Draw a vector w ⃗ parallel to ⃗v . Move w ⃗ around. What do you notice. Is ⃗v parallel to ⃗v ? Use the Denition of parallel vectors to answer. Next we will make sense of perpendicular vectors and angle between vectors in Rn . Denition (Dot Product). Let ⃗ v and w ⃗ be (row or column) vectors with components v1 , v2 , · · · , vn and w1 , w2 , · · · , wn respectively. The dot product of ⃗v and w ⃗ is a scalar denoted by ⃗v · w ⃗ and is dened as ⃗v · w ⃗ = v1 w1 + v2 w2 + · · · + vn wn . Note that ∥⃗x∥ = √ ⃗x · ⃗x. Dot product can be interpreted geometrically. With some high school geometry, we can show that for 2 3 vectors in R and R ⃗v · w ⃗ = cos θ∥⃗v ∥∥w∥, ⃗ where θ is the angle between ⃗v and w ⃗. Denition (Angle Between Vectors). Given vectors ⃗v and w⃗ , the angle between ⃗v and w⃗ in Rn is dene to be3 arccos( ⃗v · w ⃗ ). ∥⃗v ∥∥w∥ ⃗ Now we can extend our intuition about perpendicular vectors to Rn . Denition (Orthogonal Vectors). We say two vectors ⃗v and w⃗ in Rn are perpendicular or orthogonal if ⃗v · w ⃗ = 0. Exercise. (2) (3) 3this Does the denition of orthogonal vectors match your intuition in R2 and R3 ? If ⃗v and w ⃗ are orthogonal, what is the angle between them? Is there an exception to your answer? Is there a vector in Rn that is orthogonal to all other vectors un Rn ? (1) denition makes sense thanks to Cauchy Schwartz inequality that guarantees −1 ≤ ⃗ v ·w ⃗ ∥⃗ v ∥∥w∥ ⃗ ≤ 1. MAT 188-WEEK 1 Sets, Vectors, Lines, Planes 7 Lines and Planes Lines in R2 . You know from high school that the equation plane that passes through the origin and has slope to the (linear) equation R2 in y = ax. in y = ax. a. y = ax represents a line In other words, ℓ ℓ on the Cartesian represents the set of all solutions We can identify points on the Cartesian plane with (the tips) of vectors standard position. This way, ℓ can be thought of as the set of all vectors x y in R2 such that x l={ ∈ R2 | y = ax} y Let d⃗ be a nonzero vector on the line ℓ. We say ⃗ l = {⃗x ∈ R2 | ⃗x = k d, d⃗ is for some a direction vector for k ∈ R} or ℓ, and we can describe ℓ as l = {k d⃗ | k ∈ R} ⃗x = k d⃗ as the vector form or vector parametric form of the line ℓ. The scalar k is a varies over R. It is allowed to take any value so that all the points on the line ℓ can be We often refer to parameter that obtained from this equation. When we express a line in its vector form, we always think of vectors in their standard position. m R2 , a, that does not go through the origin. Such ⃗ be a nonzero described as the set of all solutions to y = ax + b, for some nonzero b. Let d and let p ⃗ be the position vector of a point p on m. Then we can describe m as Now consider a line in with slope m = {⃗x ∈ R2 | ⃗x = k d⃗ + p⃗, We refer to ⃗x = k d⃗ + p⃗ as the vector form that goes through the tip of vector p⃗ for some or k ∈ R} or lines can be vector on m m = {k d⃗ + p⃗ | k ∈ R} vector parametric form of a line with direction vector in the standard position. If we describe d d⃗ = 1 d2 , and p⃗ = p1 p2 d⃗ in terms of their components, then the vector form can be broken into an equation of each component. x1 = kd1 + p1 x2 = kd2 + p2 (0.1) That system of equations (0.1) are called the Another way to describe the line ℓ ℓ parametric form with equation y = ax is any none-zero vector perpendicular to ℓ. Pick a vector Rn that are perpendicular to ⃗n. In other words, of the line m. is via a normal vector. A normal vector for ⃗n perpendicular to ℓ. Then ℓ is the set of vectors in l = {⃗x ∈ R2 | ⃗x · ⃗n = 0}. Lines in R3 . We can describes the vector form and the parametric from of a line in R 3 exactly the same d1 2 3 ⃗ = d2 that is d⃗ is a way as we did for lines in R . Consider a line ℓ in R , with direction vector d d 3 p1 vector on line ℓthat passes that goes thorough the point p = (p1 , p2 , p3 ). Let p ⃗ = p2 be the position p3 vector of p. Then ℓ can be described as l = {⃗x ∈ R3 | ⃗x = k d⃗ + p⃗, with the vector form ⃗x = k d⃗ + p⃗ for some k ∈ R} and the parametric equation or l = {k d⃗ + p⃗ | k ∈ R} MAT 188-WEEK 1 Sets, Vectors, Lines, Planes 8 x1 = kd1 + p1 x2 = kd2 + p2 x3 = kd3 + p3 . (0.2) Exercise. Does the equation y = ax represent a line in R3 ? Lines in R2 and R3 . The only plane in R2 is the Cartesian plane R2 itself. So there is not much there 3 to look at. Let's think about a plane p in R that passes through the origin. Can we come up with a vector equation, a parametric equation and a way to describe a vector perpendicular to the plane the plane p. p? ⃗n in terms of it's normal vector, that is ⃗n p is perpendicular to ⃗n p can be described as Let's start with the latter. Let The position vector of every point on position that is perpendicular to p is on p. hence, be a vector perpendicular to and every vector in standard p = {⃗x ∈ R3 | ⃗x · ⃗n = 0}. n1 Let ⃗ n = n2 n3 and x1 ⃗x = x2 . x3 The normal equation ⃗x · ⃗n = 0 unpacks into n1 x1 + n2 x2 + n3 x3 = 0. Exercise. Let b be a non-zero scalar. Explain what the solution to n1 x1 + n2 x2 + n3 x3 = b describes R3 ? Does it pass through the origin? To describe the vector form and the parametric equation of linear combination. We will see this next week! m, we need to understand the concept of