1-2 Geometric vectors

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1 1-2 Geometric ectors We are going to start simple, by defining 2-dimensional ectors, the simplest ectors there are. Are these the ectors that can be defined by to numbers only? Yes, and here is a formal definition: Definition A 2-dimensional ector (or simply 2D ector) consists of an ordered pair of scalars a and b called components. Like speed and direction? It can be done that ay, but e shall use a different approach, one that makes use of something that you hae used umpteen times before, namely the coordinates of a point in the Cartesian -y plane. Are you saying that a ector ith components a and b is the same as a point (a, b)? I said that e shall use coordinates to gie meaning to components, not that they are the same thing. To emphasize the connection, but also to see the differences, e shall follo this conention in the notation Definition A 2D ector ith components a and b ill be denoted by a loer case bold letter and its components ill be listed in square brackets, as in: ab To denote the components of a generic ector e shall use subscripts, as in: As you can see, e use a boldface, loer case letter to represent a ector, as opposed to the italic capital letters used to represent a point: P(a, b). Also, e place the to components in square brackets instead of the round brackets used for coordinates. Finally, heneer possible e shall separate the components by a space rather than the comma that is traditional ith coordinates Eample 2 The ordered pairs 12, 4 5, 8 are all 2D ectors, as ell as any other pair of numbers you can imagine. But remember that the ector 12 is not the same as the ector 21, since the order in hich the Chapter 1: Geometric ectors Page 1 Section 2: Geometric ectors

2 numbers are listed is important in a ector, just as it is for the coordinates of a point. And keep in mind that the ector 12 is not the same as the point P (1, 2). This chapter is called Geometric ectors, yet you are saying that ectors are not points. So here is the geometry and here is the connection beteen points and coordinates? One clarification coming up: Definition A to dimensional ector ab can be represented geometrically as a directed segment, that is, an arro hose run is a and hose rise is b, as shon in the picture. We shall refer to the starting point of the arro representing a 2D ector as its tail and its end point as its tip. A 2D ector can be represented by an arro starting at any point, depending on the application, and sometimes there is no need to specify such starting point. a b Eample Here you can see a geometric representation of the 2D ectors 35 23and 3 4, all starting at the origin. u, You can also see another representation of the ector that is starting at some other point. Again remember that hile you may be used to thinking of these arros as being the ectors, you need to start thinking of them simply as geometrical or graphical representations of the corresponding ectors. u Chapter 1: Geometric ectors Page 2 Section 2: Geometric ectors

3 1.2.6 Eample The ector 2 3 can be represented by the arro starting at 1, 1 and ending at 1, 4, or by the arro starting at 5, 2 and ending at 3, 5, as shon in the picture. Once again, each of these arros is a representation of the ector, not the ector itself. The folloing technical fact, hich you hae probably seen before, is an immediate consequence of definition and you can see an illustration of it in the last eample Technical fact If the points P1 1, y1 and P2 2, y2 respectiely of a arro representing the ector In the same ay e hae that: are the tail and tip, y y 2 1, y y 2 1, then: Definition Gien to points P p, p and Q q, q, the ector PQ is the one represented by the arro ith P as tail and Q as tip: PQ q p q p 1 2 Gien a point P p, p, the ector p p PO represented by the arro ith the origin O as tail and P as tip can also be denoted by P. This oerlap of notation can generate significant simplifications in most cases, but should be aoided if it creates confusion instead. Speaking of hat e hae seen before, I remember that in preious courses ectors ere denoted ith an upper arro, like. Is that rong? Chapter 1: Geometric ectors Page 3 Section 2: Geometric ectors

4 Certainly not, but that notation reminds us of the arro idea, hich e shall consider only as one possible use of ectors. In fact there are other notations that are used in different books and by different people to represent ectors. Here is a sample of such alternaties, in case you find them somehere else Knot on your finger The folloing are acceptable and used alternatie ays of denoting a ector: Single letter symbols:,, Component notation: a b, a, b, a, b, a b So, hy don t you use them? I ill not use the upper and loer bars or the upper arro both to sae ink () and because I find them redundant hen it is clear that e are dealing ith ectors. Also, I ill not use the other component notations because they are easily confused ith other symbols (such as point coordinates) and for other reasons that ill become apparent as e progress. Hoeer feel free to use these alternatie notations and be prepared to identify and understand them hen they are used by other people. Before e moe on, here is a definition that may seem pathetically simple no, but ill be used etensiely later, both to simplify our discussions and to eliminate pedantic little problems. The ector Definition 0 is called the zero ector (duh!) and any other ector is said to be a non-zero ector. No that e hae looked at the most basic ectors, let us step up a bit and make the acquaintance of 3D ectors. I suspect that e just need to etend the concepts e hae seen so far to 3-dimensional space, right? Of course! The only problem is that the pictures become more difficult to dra on a 2-dimensional piece of paper. But the concepts are the same: Definition A 3-dimensional (3D) ector is an ordered triple of scalars, called components. Such a ector is denoted by abcor and can be represented by a directed arro in 3-dimensional space haing a, b and c as the corresponding lengths in the, y and z directions Chapter 1: Geometric ectors Page 4 Section 2: Geometric ectors

5 respectiely. All terminology related to the arro representation of 2D ectors etends to 3D ectors, including the concepts of zero and non-zero ectors. Z Here is a representation of a 3D ector abc anchored at the origin. Keep in mind that because of problems ith perspectie, it is difficult to dra and to see a 3D ector that is not anchored at the origin, but fortunately, e shall not hae to do this often. b c a Y Hmmm, I can see hy the pictures are more difficult to isualize and to dra! So, as much as possible e ll stick to 2D ectors for illustration and leae the rest to your isual imagination. The folloing technical fact is the obious etension of Fact 1.2.7, but I am repeating it here to emphasize its eistence and usefulness Technical fact If the points P, y, z and P, y, z are the tail and tip respectiely of a geometrical representation of the ector, then: In the same ay:, y y, z z , y y, z z Definition Gien to points P p, p, p and Q q, q, q, the ector Chapter 1: Geometric ectors Page 5 Section 2: Geometric ectors

6 PQ is the one represented by the arro ith P as tail and Q as tip: PQ q p q p q p Gien a point P p, p, p, the ector PO p p p represented by the arro ith the origin O as tail and P as tip can also be denoted by P. This oerlap of notation can generate significant simplifications in most cases, but should be aoided if it creates confusion instead. Reie questions: Checkpoint for Section 1-2: Geometric ectors 1. What is a 3D ector? 2. What are the components of a 3D ector? 3. When e identify a point ith a ector in R 3, hat do e assume about the ector? 4. Present some alternatie notations commonly used to indicate the components of a 3D ector abc. Theory questions: 5. What is a 2D ector? 6. What is the name of the scalars that together constitute a ector? 7. What are the components of a 2D ector? 8. Do the pairs [] and [2 1] represent the same ector? 9. What is the difference beteen the ector [] and the point (1, 2)? 10. Is a 2D ector a directed segment? 11. What is the tip of an anchored ector? 12. What does an arro representing the zero ector look like? 13. Do the pairs [] and [2 1 3] represent the same ector? 14. What is the difference beteen the ector [] and the point (1, 2, 3)? 15. Is a 3D ector a directed segment? Computation questions: 16. Dra an arro representing each of the folloing ectors: 7 5 a) u 25 b) 1 c) 4 2 y Then dra another arro representing the same ectors, but ith a different tail. d) Chapter 1: Geometric ectors Page 6 Section 2: Geometric ectors

7 17. Dra an arro representing each of the folloing ectors: a) [-2 5-1] b) [1-1] c) d) [ ] Then dra another arro representing the ectors, but ith a different tail. 18. Determine the tail of the geometric representation of the ector hose tip is at 1, 6, 2. Proof questions: 19. Eplain ho to use Definition to proe Fact Templated questions In these questions, make your on choice of the ector, either 2D or 3D, and the points P and Q. 20. Determine the tip of the arro representing the ector ith the tail at P. 21. Determine the tail of the arro representing the ector ith the tip at P. 22. Determine the ector corresponding to the arro tail at P and tip Q. and your questions are Chapter 1: Geometric ectors Page 7 Section 2: Geometric ectors