Chapter 1: Introduction to Vectors (based on Ian Doust s notes)

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1 Chapter 1: Introduction to Vectors (based on Ian Doust s notes) Daniel Chan UNSW Semester Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

2 A typical problem Chewie points his crossbow south-east. If bolts fly at 5ms 1, it s easy to determine where it is at any point in time. Question But what if he fires it from the Millenium Falcon which is moving north at 10ms 1? Pictorial Ans Q What if the Millenium Falcon is in the Death Star which is moving...? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

3 Goals of this chapter Note that to answer the question above, you need to know both the magnitudes 5ms 1, 10ms 1 and the directions of motion. In this chapter we ll intro new mathematical objects called vectors which encode info about magnitudes & directions. Note real numbers only encode magnitudes and sign (i.e. + or ). study how vectors combine to answer questions such as that above. intro coordinates to reduce vector calculations to calculations involving numbers. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

4 What s a (geometric) vector? A (geometric) vector is often represented by an arrow or a directed line segment. It can represent a velocity like 5ms 1 south east. The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector. Typical notation for a vector: v, v, or ṽ. We denote the magnitude of v by v. (No notation for the direction!) Equality of vectors We say vectors u & v are equal, and write u = v, if u and v have the same magnitude and direction. E.g. The zero vector, denoted by 0, has length 0. It is the only vector with no specific direction. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

5 Vector addition Given two vectors u and v we can add them head-to-tail to produce a new vector u + v as follows: Adding the zero vector 0 does nothing as v + 0 = 0 + v = v for all vectors v. There are many physical interpretations of this addition. E.g. 3-way tug-o-war We let v denote the vector with the same magnitude but the opposite direction so v + ( v) = 0. We define subtraction by u v := u + ( v). Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

6 Reminder on real numbers Primary and high school arithmetic uses numbers that we call real. For example, the numbers 0, 73, 2 1 5, 2, π e 2,.... and so on are real numbers. We will denote the real number system by R. We visualize R as an infinitely long line: Within the set R we have Natural numbers, N = {0, 1, 2, 3,...}. Integers, Z = {..., 2, 1, 0, 1, 2,...}. { Rational numbers, Q = p q }. : p, q Z, q 0 Positive numbers, R + = {x R : x > 0}. Irrational numbers, i.e. those that aren t in Q. and many other subsets of numbers. Sometimes, we refer to numbers as scalars (as opposed to vectors). Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

7 Scalar multiplication There is another operation we can perform on vectors, scalar multiplication. If λ R is a non-negative scalar and v a vector, then λv denotes the vector with the same direction, but magnitude is scaled by λ, i.e. λv = λ v. If λ < 0 then we define λv := λ ( v). E.g. ( 1)v =?? If u = λv, then we say that u and v are parallel. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

8 Properties of vector addition Question In what sense is vector addition a type of addition? A You can manipulate vector sums much as you can numbers because of the Commutative law v + w = w + v (Why?) Associative law (u + v) + w = u + (v + w) (Why?) Since these sums equal, we ll write it simply as u + v + w. Similarly, we may omit brackets when adding 4 or more vectors together. Challenge Q Why? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

9 Properties of scalar multiplication Question In what sense is scalar multiplication a type of multiplication? Associative law λ(µv) = (λµ)v (Why?) Distributive law λ(v + w) = λv + λw (Vector) (λ + µ)v = λv + µv (Scalar) Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

10 Vector arithmetic E.g. Let w = u + 2v. Are 2w 4v & u parallel? A We simplify 2w 4v = 2(u + 2v) 4v Note To perform this arithmetic, we only needed the basic properties of vector addition and scalar multiplication above. In general, we will meet lots of contexts where we have a vector addition and scalar multiplication satisfying these axioms. These sets will be called vector spaces. In these cases, we can perform arithmetic as above. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

11 Co-ordinates To put co-ordinates on the plane we need to specify: an origin point O where (co-ord axes cross), and a pair of vectors i and j which have length 1 and which are at right angles to one another. (The convention is to choose j at π/2 anticlockwise from i.) So i, j give direction of coord axes & scale. Fact-Definition Every geometric vector a in the plane can be written as a = a 1 i + a 2 j for some unique pair of numbers a 1, a 2 R. ( ) a1 The co-ordinates or coordinate vector of a (with respect to i, j) is. a 2 Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

12 Co-ordinate arithmetic reflects vector arithmetic Let a = a 1 i + a 2 j, b = b 1 i + b 2 j so coords are The coords of ( a1 a 2 ), ( b1 b 2 ). a ± b = (a 1 i + a 2 j) ± (b 1 i + b 2 j) = (a 1 ± b 1 )i + (a 2 ± b 2 )j ( ) a1 ± b are 1. a 2 ± b 2 Sim, the coords of ( ) λa1 are. λa 2 λa = λ(a 1 i + a 2 j) = λa 1 i + λa 2 j Upshot This says to sum, subtract or multiply vectors, we need only sum, subtract or multiply coords. Challenge Q How do coords change if you replace i j, j i? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

13 Example To find coords recall given a right angle triangle, Question I walk 1km due west, then 4km on a bearing 30 east of north. Where do I end up? Solution. Take i pointing east and j pointing north & units are km. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

14 3-dimensional version You can do all this with 3-dimensional geometric vectors too. Here you need basis vectors i, j and k. You can write vectors in form which have coords Again a = a 1 i + a 2 j + a 3 k, a 1 a 2 a 3, b 1 b 2 b 3. b = b 1 i + b 2 j + b 3 k a 1 + b 1 λa 1 coords of a + b = a 2 + b 2, coords of λa = λa 2. a 3 + b 3 λa 3 Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

15 The space R n We can generalise coordinate vectors to any number of components! Let n be a positive integer. An n-tuple or n-vector is an ordered list of n numbers a 1, a 2,..., a n, written as either a column vector or (less often in this course) a row vector: a 1 a 2 a =. or a = (a 1, a 2,..., a n ). a n The set of all n-tuples is denoted R n. Thus ( 0 R 0) 2, 1 2 R 3, R4. π Hence coord vectors of 3-dim vectors lie in R 3 whilst those of 2-dim vectors lie in R 2. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

16 Arithmetic on R n We can define vector addition and scalar multiplication on R n coordinatewise as we saw for coordinate vectors: a 1 b 1 a 1 + b 1 a 1 λa 1 a 2. + b 2. := a 2 + b 2., λ a 2. := λa 2.. a n b n a n + a n a n λa n 0 a 1 a 1 The zero element is 0 =. and the negative is given by. =. 0 a n a n (so we can define subtraction!). Note: each R n is a separate system. You can t add a 3-tuple to a 7-tuple! 1 1 Q Compute = 2 2 Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

17 Properties of arithmetic on R n This vector addition and scalar multiplication inherit good properties from addition and multiplication on R. That is, if a, b, c R n and λ, µ R then Commutative: a + b = b + a, Associative: (a + b) + c = a + (b + c), Distributive: λ(a + b) = λa + λb, Distributive: (λ + µ)a = λa + µa. Cancellation: λa = 0 if and only if λ = 0 or a = 0, etc Moral: You can do algebra in R n without running into any problems! Proof Easy from definitions but take up space e.g. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

18 Displacement vector Put coords on the plane by specifying O, i, j as usual. Given any 2 points A, B on the plane, we define the displacement vector AB to be the geometric vector with tail A & head B i.e. From the picture we see AB = OB OA. Important Remark In high school you would have considered coords of the point A (as opposed to a vector). coords A = coords OA. coords AB = coords B - coords A. OA is called the position vector of A (with respect to O). These observations also hold if the points are in space with coords specified. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

19 Simple geometric applications of vectors E.g. Are A = 1, B = 0, C = 2 collinear? E.g. Are A = parallelogram? ( ( ( ( , B =, C =, D = the vertices of a 1) 3) 5) 3) Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

20 Length and distance in R n ) Pythagoras thm = the length of a geometric vector with coords is a 2 a a2 2. This suggests the following generalisation of the length concept to Rn. Definition Let a, b R n. The length of a is defined to be a = a a2 n. ( a1 the distance between a and b is defined to be dist(a, b) = b a. 1 Example. a) What is 1 1? 1 b) Suppose that the point A has coordinates (1, 2, 3) and the point B has coordinates ( 1, 2, 5). What is the distance between A and B? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

21 Lines in R n Using our 2 and 3-dim intuition, a line L in R n should be determined by a point A on the line, say with a = OA and, a direction, say given by a non-zero vector v Let s determine what the general point of L ought to be: x = a + λv, λ R This is called the parametric vector form of the line L. We call λ the parameter, and as it varies over R, the variable x varies over all the points of the line L. Definition A line in R n is any set of the form {x R n x = a + λv, λ R} for some fixed vectors 0 v, a R n. Note a gives a point on the line and v its direction. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

22 Vectors and MAPLE See MAPLE file Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

23 Midpoints Midpoints Let a, b R n be the position vectors for the points A, B. The midpoint of AB has coordinates OA + 1 AB = a (b a) = 1 (a + b). 2 Why? Q Show that the diagonals of a parallelogram bisect each other. Challenge Q What s the 3-dim version of this result? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

24 Finding parametric forms for lines from Cartesian form In high school, you express a line in the plane in Cartesian form ax + by = c. Question Find a parametric vector form for the line y = 3x + 2 in R 2. A1 Consider 2 points on the line A2 We introduce the parameter λ = N.B. There are many other solutions! (What are they?) Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

25 Parametric to Cartesian form Question Write the line x = ( ( λ, λ R in Cartesian form. 1) 1) The secret is to eliminate ( ) ( the ) extra variable ( ) ( λ! ) x λ Solution. Write = + λ =. Then y λ λ = What about x = ( ( λ, λ R? 1) 1) Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

26 Cartesian form for lines and planes in R 3 Recall that a plane in xyz-space can be described by an equation ax + by + cz = d where not all a, b, c are 0. This is called the Cartesian form for the plane. The terms in this equation can of course be re-arranged many ways (see below). To obtain the cartesian form for a line L, we need 2 such equations. Each defines a plane P 1, P 2 and solving simultaneously gives the solution P 1 P 2. This will be a line unless.... Usually, (but not always) we can write the 2 equations in the form for some constants a i, v i. x a 1 v 1 = y a 2 v 2 = z a 3 v 3 Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

27 Cartesian to parametric form for lines in R 3 Question Find the parametric form for x 2 3 = y = z 3 2. A We can find a point on the line and a direction vector or just introduce the parameter Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

28 Parametric to cartesian form for lines in R 3 Question Find a cartesian form for x = λ 4 5, 6 λ R 1 4 What about x = 2 + λ 5? 3 0 Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

29 Parametric vs Cartesian form The Cartesian form expresses a line or plane as the solutions to some equations. (Top down) The parametric form expresses the line or plane by a sophisticated way of listing elements, where running through the list is by letting the parameter range over a set.(bottom up) In mathematics, these are the two usual general ways to describe any set. Both have their uses. For lines, the parametric form is closest to our geometric picture of the line. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

30 Why bother defining lines in R n? Suppose we are solving equations in n unknowns x 1,..., x n. If n = 3, it is often good to visualise the solution set in x 1 x 2 x 3 -space. For example, solving simultaneously a 1 x 1 + a 2 x 2 + a 3 x 3 = a, b 1 x 1 + b 2 x 2 + b 3 x 3 = b should on geometric grounds, give either a line, plane or the empty set. In particular, you can t get a point or two points etc. If n > 3, we can use our geometric intuition to understand solutions to many equations provided we generalise our notions of things like lines in R 3 to lines in higher dimensions. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

31 Directions of a plane Thought experiment What are the directions of a plane P R 3? Since we are interested in directions only, let s suppose P passes through O and that v, w P are not parallel Hence Fact Any vector parallel to P has the form λv + µw for some λ, µ R. i.e. all other directions of P can be obtained from v, w by combining them using vector operations. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

32 Linear combination More generally, we consider Definition Suppose that v 1, v 2,..., v k R n. A linear combination of these vectors is a vector of the form λ 1 v 1 + λ 2 v λ k v k with λ 1,..., λ k R. Q Is 1 2 a linear combination of 1 0 and 2 1? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

33 Span Definition Let v 1, v 2,..., v k R n. The span of v 1, v 2,..., v k, written span(v 1,..., v k ) is the set of all linear combinations of v 1, v 2,..., v k. i.e. span(v 1,..., v k ) = {λ 1 v 1 + λ 2 v λ k v k λ 1,..., λ k R}. ( 2 Ex. Describe span. 1) More gen, span(v) is the 1 0 Ex. Describe span 0, Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

34 Planes in R n Definition A plane in R n is defined to be a set of the form S = {a + λ 1 v 1 + λ 2 v 2 λ 1, λ 2 R}, where a, v 1 and v 2 are fixed vectors in R n, and v 1 and v 2 are not parallel. The expression x = a + λ 1 v 1 + λ 2 v 2, λ 1, λ 2 R is a parametric vector form for the plane through a parallel to the vectors v 1 and v 2. The above picture shows that when n = 3, our definition agrees with our old one. Q What if v 1, v 2 above are parallel? Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

35 Parametric form for plane determined by 3 points Question Find a parametric vector equation for the plane through the points a = 2 0 b = 1 and c = , 1 Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

36 Finding Cartesian form for planes from parametric form Question Find the Cartesian equation of the plane in R 3 x y = λ λ 2 3 2, λ 1, λ 2 R. z Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

37 Meanwhile back at the Death Star Question 1 2 The Millenium Falcon, at coords 0 is flying in direction 1. Will it hit the 1 0 Death Star wall, a plane with Cartesian eqn x y z = 1? A Without the Death Star, the flight trajectory would be the ray with parametric equation Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38

38 Standard basis vectors for R n In R n, the vector e j is the n-tuple with 1 in the jth position and zeros elsewhere. ( ( R : e 1 =, e 0) 2 =. 1) R 3 : e 1 = 0, e 2 = 1, e 3 = Obviously, every vector in R n can be written uniquely as a linear combination of e 1,..., e n, eg x 1 x 2 = x x = x 1 e 1 + x 2 e 2 + x 3 e 3. x x 3 The vectors e 1,..., e n are called the standard basis vectors for R n. Daniel Chan (UNSW) Chapter 1: Introduction to Vectors Semester / 38