Mathematics
Vector Addition
Vector addition is the process of combining two or more vectors to produce a new vector. This is done by adding the corresponding components of the vectors together. Geometrically, vector addition can be visualized as placing the tail of one vector at the head of another and drawing a new vector from the tail of the first to the head of the second.
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11 Key excerpts on "Vector Addition"
- Gregory J. Gbur(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
For most of this chapter we reserve the term “vector” for quantities which possess magnitude and direction in three-dimensional space, and are independent of the specific choice of coordinate system in a manner to be discussed in 1 2 Vector algebra A A B B C Figure 1.1 The parallelogram law of Vector Addition. Adding B to A (the addition above the C-line) is equivalent to adding A to B (the addition below the C-line). Section 1.2. We briefly describe vector spaces at the end of this chapter, in Section 1.5. The interested reader can also consult Ref. [Kre78, Sec. 2.1]. Vector Addition is commutative and associative; commutativity refers to the observation that the addition of vectors is order independent, i.e. A + B = B + A = C. (1.1) This can be depicted graphically by the parallelogram law of Vector Addition, illustrated in Fig. 1.1. A pair of vectors are added “tip-to-tail”; that is, the second vector is added to the first by putting its tail at the end of the tip of the first vector. The resultant vector is found by drawing an arrow from the origin of the first vector to the tip of the second vector. Associativity refers to the observation that the addition of multiple vectors is independent of the way the vectors are grouped for addition, i.e. (A + B) + C = A + (B + C). (1.2) This may also be demonstrated graphically if we first define the following Vector Additions: E ≡ A + B, (1.3) D ≡ E + C, (1.4) F ≡ B + C. (1.5) The vectors and their additions are illustrated in Fig. 1.2. It can be immediately seen that E + C = A + F. (1.6) So far, we have introduced vectors as purely geometrical objects which are independent of any specific coordinate system. Intuitively, this is an obvious requirement: where I am standing in a room (my “position vector”) is independent of whether I choose to describe it by measuring it from the rear left corner of the room or the front right corner.
eBook - ePub- L.R. Shorter(Author)
- 2014(Publication Date)
- Dover Publications(Publisher)
CHAPTER I
ADDITION
[1]. Definition of a Vector.
A vector is a magnitude which can be represented by a straight line of finite length drawn in a definite direction.In the following pages the term vector will be applied both to the magnitude itself and to its representation by a finite straight line drawn in a parallel direction.Nothing is said in this definition of the point from which the vector starts—the initial point—so that we may consider the position of this point to be arbitrary.There are, then, an infinite number of vectors having the common properties of a given finite length and a given direction. Hence all vectors having the same given length and the same direction are to be considered equivalent: thus a vector from any initial point may be replaced by an equivalent vector from any other initial point. The point which terminates a vector is called its terminal point or extremity.Vectors in the text will be represented by clarendon letters, such as a, b, c, r; but when referred to in diagrams, they will be represented by large capitals in clarendon, thus AB, CD.Finite algebraical quantities in which direction is not involved are called scalar quantities or scalars: these will be represented in the text in ordinary type.Such quantities as Force, Acceleration, Velocity (when the direction of motion of the latter is taken into account), Fluid Flow in a definite direction, are examples of vectors.Among scalars are included: the temperature of a body at a given point in it; the work done by forces on a body when it is displaced by their action from one position to another; the mass of a body; an interval of time.[2]. Addition of Vectors.
The term “addition” is applied to a method of combining vectors, which is analogous to the ordinary addition of algebraical quantities, and which is familiar in the graphic composition and resolution of forces.FIG . 1.Let AB represent a vector, with initial point A and terminal point B (Fig. 1 ), and let another BC be drawn from the terminal point B of AB. The plus sign in clarendon type, +, will be employed to denote this operation. Calling the vector AB, a and the vector BC, b, the process just described is symbolised by a + b.
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
That is, (2.3.2) FIGURE 2.2.1 A fixed line L and a fixed vector V . FIGURE 2.3.1 Two vectors A and B to be added. FIGURE 2.3.2 Vectors A and B connected head to tail. O Y L Z X n n n z y x V R A B = + A B B A + = + A B A B Review of Vector Algebra 17 Vector subtraction may be defined from Vector Addition. Specifically, the difference of two vectors A and B , written as A – B , is simply the sum of A with the negative of B . That is, (2.3.3) An item of interest in Vector Addition is the magnitude of the resultant, which may be determined using the geometry of the parallelogram and the law of cosines. For example, in Figure 2.3.5, let θ be the angle between A and B, as shown. Then, the magnitude of the resultant R is given by: (2.3.4) Example 2.3.1: Resultant Magnitude To illustrate the use of Eq. (2.3.4), suppose the magnitude of A is 15 N, the magnitude of B is 12 N, and the angle θ between A and B is 60˚. Then, the magnitude of the resultant R is: (2.3.5) Observe from Eq. (2.3.4) that if we double the magnitude of both A and B , the magnitude of the resultant R is also doubled. Indeed, if we multiply A and B by any scalar s , R will also be multiplied by s . That is, (2.3.6) This means that Vector Addition is distributive with respect to scalar multiplication. Next, suppose we have three vectors A , B , and C , and suppose we wish to find their resultant. Suppose further that the vectors are not parallel to the same plane, as, for example, in Figure 2.3.6. The resultant R is obtained in the same manner as before. That is, the vectors are connected head to tail, as depicted in Figure 2.3.7. Then, the resultant R is obtained by connecting the tail of the first vector A to the head of the third vector C as in Figure 2.3.7. That is, (2.3.7) FIGURE 2.3.3 Resultant (sum) R of vectors A and B . FIGURE 2.3.4 Two ways of adding vectors A and B . FIGURE 2.3.5 Vector triangle geometry.
eBook - PDFMathematics for Engineering, Technology and Computing Science
The Commonwealth and International Library: Electrical Engineering Division
- Hedley G. Martin, N. Hiller(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
a+b a b FIG. 8.5. Thus Vector Addition is defined in accordance with the familiar parallelogram law of forces and is consistent with the way in which displacements are compounded. Some directed quanti-ties do not conform with this law and therefore are not defin-able as vectors. It follows from the definition that vector 300 MATHEMATICS FOR ENGINEERING AND TECHNOLOGY addition is commutative and associative, that is a+b = b+a (3.1) and (a+b)+c = a+(b+c). (3.2) The difference of two vectors a and b, denoted by a —b, is defined as the sum a + (— b) and is illustrated in Fig. 8.6. By the definitions of vector equality and addition a vector a is given in terms of its vector components by a = a x i + ayj + a r k, (3.3) FIG. 8.6. as shown in Fig. 8.7. The radius vector r is given by r = xi + yj + zk. (3.4) The sum of two vectors is derived by adding their respective vector components. a + b = (ax+b,)i+-(a.v+b.v)j+(a2+b,)k• (3.5) FIG. 8.7. VECTOR ANALYSIS 301 Let the direction of a vector a make angles a, b, y with the positive directions of the x, y, z axes respectively. The cosines of these angles are called the direction cosines of a and are given by cos a _ a, x /a, cos = a/a, cos y = a Z /a ; (3.6) by (2.2) the sum of their squares is unity : cost oi + cost 1 3 + cost y = 1. (3.7) If a is a unit vector then a = 1 and the direction cosines are equal to the components of á. Let a vector a make an angle Q with a vector b. the quantity a b = a cos Q is called the component of a in the direction of b and is the orthogonal projection of a on the line along b, signed according to the relative senses of a and b. The compo-nent a b is now expressed in terms of the components of a. Fio. 8.8. Figure 8.8 shows the vector components of a, parallel to the coordinate axes, and vector b with the angles a, b, y which its direction makes with the axes. Clearly the components of ai, a y j and a Z k in the direction of b are a x cos a, a y cos ß and
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Vectors, such as displacement, have both magnitude and direction (5 m, north) and obey the rules of vector algebra. ● Two vectors a → and b → may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum s → . To subtract b → from a → , reverse the direction of b → to get ‒ b → ; then add ‒ b → to a → . Vector Addition is commutative and obeys the associative law. ● The (scalar) components a x and a y of any two-dimensional vector a → along the coordinate axes are found by dropping perpendicular lines from the ends of a → onto the coordinate axes. The components are given by a x = a cos θ and a y = a sin θ, where θ is the angle between the positive direction of the x axis and the direction of a → . The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation of the vector a → with a = √a 2 x + a 2 y and tan θ = ay a x . Key Ideas Learning Objectives After reading this module, you should be able to . . . 3-1 VECTORS AND THEIR COMPONENTS 41 A vector has magnitude as well as direction, and vectors follow certain (vector) rules of combination, which we examine in this chapter. A vector quantity is a quantity that has both a magnitude and a direction and thus can be represented with a vector. Some physical quantities that are vector quantities are displacement, velocity, and acceleration. You will see many more throughout this book, so learning the rules of vector combination now will help you greatly in later chapters. Not all physical quantities involve a direction. Temperature, pressure, energy, mass, and time, for example, do not “point” in the spatial sense. We call such quantities scalars, and we deal with them by the rules of ordinary algebra. A single value, with a sign (as in a temperature of −40°F), specifies a scalar.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Vectors, such as displacement, have both magnitude and direction (5 m, north) and obey the rules of vector algebra. ● Two vectors a → and b → may be added geometrically by drawing them to a common scale and placing them head to tail. The vector connecting the tail of the first to the head of the second is the vector sum s → . To subtract b → from a → , reverse the direction of b → to get ‒ b → ; then add ‒ b → to a → . Vector Addition is commutative and obeys the associative law. ● The (scalar) components a x and a y of any two-dimensional vector a → along the coordinate axes are found by dropping perpendicular lines from the ends of a → onto the coordinate axes. The components are given by a x = a cos θ and a y = a sin θ, where θ is the angle between the positive direction of the x axis and the direction of a → . The algebraic sign of a component indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation of the vector a → with a = √a 2 x + a 2 y and tan θ = ay a x . Key Ideas Learning Objectives After reading this module, you should be able to . . . 3-1 VECTORS AND THEIR COMPONENTS 35 A vector has magnitude as well as direction, and vectors follow certain (vector) rules of combination, which we examine in this chapter. A vector quantity is a quantity that has both a magnitude and a direction and thus can be represented with a vector. Some physical quantities that are vector quantities are displacement, velocity, and acceleration. You will see many more throughout this book, so learning the rules of vector combination now will help you greatly in later chapters. Not all physical quantities involve a direction. Temperature, pressure, energy, mass, and time, for example, do not “point” in the spatial sense. We call such quantities scalars, and we deal with them by the rules of ordinary algebra. A single value, with a sign (as in a temperature of −40°F), specifies a scalar.
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Figure 11 shows how the analytic definition of addition corresponds to the geometric one. ALGEBRAIC OPERATIONS ON VECTORS If u 8 a 1 , a 2 9 and v 8 b 1 , b 2 9 , then u v 8 a 1 b 1 , a 2 b 2 9 u v 8 a 1 b 1 , a 2 b 2 9 c u 8 ca 1 , ca 2 9 c [ R x y 2 0 4 w w w w FIGURE 9 x y a/ a¤ v= a/, a¤ H11022 H11021 | v |=oe∑∑∑∑∑∑ a™ /+a™¤ 0 FIGURE 10 u v u+v b¤ a¤ b/ a/ FIGURE 11 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 9.1 ■ Vectors in Two Dimensions 633 EXAMPLE 3 ■ Operations with Vectors If u 2, 3 and v 1, 2, find u v, u v, 2 u, 3 v, and 2 u 3 v. SOLUTION By the definitions of the vector operations we have u v 8 2, 39 8 1, 29 8 1, 19 u v 8 2, 39 8 1, 29 8 3, 59 2 u 28 2, 39 8 4, 69 3 v 38 1, 29 8 3, 69 2 u 3 v 28 2, 39 38 1, 29 8 4, 69 8 3, 69 8 1, 09 Now Try Exercise 31 ■ The following properties for vector operations can be easily proved from the definitions. The zero vector is the vector 0 0, 0. It plays the same role for addition of vectors as the number 0 does for addition of real numbers. PROPERTIES OF VECTORS Vector Addition Multiplication by a scalar u v v u c 1 u v 2 c u c v u 1 v w 2 1 u v 2 w 1 c d 2 u c u d u u 0 u 1 cd 2 u c 1 d u 2 d1 c u 2 u 1 u 2 0 1 u u Length of a vector 0 u 0 0 c u 0 0 c 0 0 u 0 c 0 0 A vector of length 1 is called a unit vector. For instance, in Example 2(c) the vector w 3 5 , 4 5 is a unit vector.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
We make no distinction between an arrowhead at the end of the line and one along the line’s length, but rather use that which gives the clearer diagram. (d) magnetic field, (e) fluid velocity component, (f) pressure, (g) electric current, (h) potential energy, (i) height, (j) gradient, (k) voltage, (l) surface charge density, (m) pressure gradient. 9.2 Addition, subtraction and multiplication of vectors • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • The resultant or vector sum of two displacement vectors is the displacement vector that results from performing first one and then the other displacement, as shown in Figure 9.1 ; this process is known as Vector Addition. However, the principle of addition has physical meaning for vector quantities other than displacements; for example, if two forces act on the same body then the resultant force acting on the body is the vector sum of the two. The addition of vectors only makes physical sense if they are of a like kind, for example if they are both forces acting in three dimensions. It may be seen from Figure 9.1 that Vector Addition is commutative, i.e. a + b = b + a . (9.1) The generalisation of this procedure to the addition of three (or more) vectors is clear and leads to the associativity property of addition (see Figure 9.2 ), e.g. a + ( b + c ) = ( a + b ) + c . (9.2) Thus, it is immaterial in what order any number of vectors are added; their resultant is always the same, whatever the order of addition. The subtraction of two vectors is very similar to their addition (see Figure 9.3 ); that is, a − b = a + ( − b ) 333 9.2 Addition, subtraction and multiplication of vectors a a a b b b c c c a + ( b + c ) ( a + b ) + c b + c b + c a + b a + b Figure 9.2 Addition of three vectors showing the associativity relation. -b b a a a -b Figure 9.3 Subtraction of two vectors. a a λ Figure 9.4 Scalar multiplication of a vector (for λ > 1).
- Jeremy Dunning-Davies(Author)
- 2003(Publication Date)
- Woodhead Publishing(Publisher)
The magnitude and direction of the directed line segment are defined to be the magnitude and direction of the vector. The magnitude of the vector a is denoted by a or sometimes |a|. This magnitude is a positive real number, being the length of a directed line segment, and is 166 Vector Algebra [Ch. 6 called the modulus of the vector sometimes. A vector in the same direction as a but of unit magnitude is denoted by a. Two vectors are said to be equal if they have the same magnitude and the same direction. Symbolically, a = b if and only if a = b and a = b . All vectors represented by a zero directed line segment are defined equal to the zero vector and are denoted by 0. The zero vector has zero magnitude and is associated usually with the absence of some particular physical property. 6.3 ADDITION OF VECTORS The sum, AB* + BC*, of the two direct line segments AB* and BC* is defined to be the directed line segment AC* (see Fig. 6.3). Β Fig. 6.3 This addition satisfies the associative law, that is AB* + (BC* + CD*) = (AB* + BC*) + CD* The proof follows immediately from the above definition: A l + ( B C * + CTJ)=AB*+BD* = AD* = AC* + CD* = (AB* + BC*) + CD*. Now, if two vectors a and b are represented by the directed line segments AB* and BC* , then the sum a + b of the vectors is defined to be that vector repre-sented by the directed line segment AC*. Consider the parallelogram ABCD, as shown in Fig. 6.4, where the di-rected line segments AB* and BC* do represent vectors a and b. Sec. 6.3] Addition of Vectors 157 Β D Fig. 6 .4 It is seen that the vector b could be represented equally well by the directed line segment AD* , and the sum a + b is represented then by the diagonal of the parallelogram.
eBook - PDFFunctions Modeling Change
A Preparation for Calculus
- Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Commutativity of addition: + = + 2. Associativity of addition: ( + ) + = + ( + ) 3. Associativity of scalar multiplication: ( ) = () 4. Distributivity of scalar multiplication: ( + ) = + and ( + ) = + 5. Identities: + 0 = and 1 ⋅ = These properties are analogous to the corresponding properties for addition and multiplication of numbers. In other words, Vector Addition and scalar multiplication of vectors behave as expected. Summary for Section 12.1 • A vector is often represented by an arrow. A vector has magnitude (the length of the arrow) and direction (which way the arrow points). 12.1 VECTORS 449 • We denote the magnitude of a vector by ‖ ‖. • A scalar is another name for an ordinary number. • Addition of vectors: The sum + is graphically represented by the arrow from the tail of to the head of . • Subtraction of vectors: The difference − is graphically represented by the arrow from the head of to the head of . • The scalar multiple of a vector, where is a scalar, is determined as follows: ∙ If > 0, then points in the same direction as and is times as long. ∙ If < 0, then points in the opposite direction as and is times as long. ∙ If = 0, then = 0 , the zero vector. Exercises and Problems for Section 12.1 Skill Refresher In Exercises S1–S4, find . S1. 50 ◦ 60 ◦ S2. 113 ◦ 35 ◦ S3. 40 ◦ 65 ◦ S4. 40 ◦ In Exercises S5–S7, find . S5. 3 5 S6. 130 ◦ 3 2 S7. 55 ◦ 65 ◦ 4 EXERCISES We use scalars to describe quantities like weight because these require only a single number (for example, 50 lbs). We use vectors to describe quantities like velocity because these require more than one number (for example, 20 mph at a heading of 80 ◦ ). In Exercises 1–8, should we describe the given quantity with a vector or a scalar? 1.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
(3.1.3) Components of a Vector The (scalar) components a x and a y of any two-dimensional vector a → along the coordinate axes are found by dropping perpendicular lines from the ends of a → onto the coordinate axes. The components are given by a x = a cos θ and a y = a sin θ, (3.1.5) where θ is the angle between the positive direction of the x axis and the direction of a → . The algebraic sign of a compo- nent indicates its direction along the associated axis. Given its components, we can find the magnitude and orientation (direction) of the vector a → by using a = √ _ a x 2 + a y 2 and tan θ = a y _ a x . (3.1.6) Unit‑Vector Notation Unit vectors ˆ i, ˆ j, and k ̂ have magnitudes of unity and are directed in the positive directions of the x, y, REVIEW & SUMMARY and z axes, respectively, in a right-handed coordinate system (as defined by the vector products of the unit vectors). We can write a vector a → in terms of unit vectors as a → = a x ˆ i + a y ˆ j + a z k ̂ , (3.2.1) in which a x ˆ i, a y ˆ j, and a z k ̂ are the vector components of a → and a x , a y , and a z are its scalar components. Adding Vectors in Component Form To add vectors in com- ponent form, we use the rules r x = a x + b x r y = a y + b y r z = a z + b z . (3.2.4 to 3.2.6) Here a → and b → are the vectors to be added, and r → is the vector sum. Note that we add components axis by axis. We can then express the sum in unit-vector notation or magnitude-angle notation. Product of a Scalar and a Vector The product of a scalar s and a vector v → is a new vector whose magnitude is sv and whose direction is the same as that of v → if s is positive, and opposite that of v → if s is negative. (The negative sign reverses the vector.) To divide v → by s, multiply v → by 1/s. The Scalar Product The scalar (or dot) product of two vectors a → and b → is written a → ⋅ b → and is the scalar quantity given by a → ⋅ b → = ab cos ϕ, (3.3.1) in which ϕ is the angle between the directions of a → and b → .
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