DOCSLIB.ORG

Vectors and 2D Motion Vectors and Scalars Vector Notation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Vectors and 2D Motion Vectors and Scalars Vector Notation Vectors and 2D Motion • Vectors and Scalars • Vector arithmetic • Vector description of 2D motion • Projectile Motion • Relative Motion -- Reference Frames Vectors and Scalars Scalar quantities: require magnitude & unit for complete description Examples: mass, tim e, temperature, speed, …… (what others?) 2.7 kg 57 °C 60 m/s Vector quantities: require magnitude, unit & direction for complete description Examples: displacement, velocity, acceleration …… (what others?) 500 m north 50 m/s heading 040° 9.8 m/s2 down Vector Notation Vector quantities are graphically represented as arrows ……. y A The length of the arrow represents the magnitude of the vector and the direction is self-evident. θ x Vectors quantities are referred to by symbol, such as… “A” or “A” (arrow used on blackboard, boldface in the text, and on overheads) A simple, unbold, unarrowed “A” refers to the magnitude of vector A. A = |A| 1 Vector Math Vectors can be multiplied by scalars: -B 3B B 0.75B The result is that the magnitude of the vector changes, but not the direction, except in the case of muliplication by a negative number where the vector reverses direction. Adding Vectors (parallelogram method) R A B A + B = R Adding Vectors ( “tip-to-tail” method) R B A A + B = R 2 Adding Vectors ( “tip-to-tail” method) R The result is A independent of the order of addition! B + A = A + B B B + A = R Subtracting Vectors B -B A A R R A + B = R A = R - B R A B Vector Components y A Ay θ x Ax Vector A (with magnitude, A) is directed at an angle θ above the +x axis A = Ax + Ay Ax: vector component of A in the x-direction Ay : vector component of A in the y-direction 3 Scalar Components y A Ay θ x Ax Given A, θ, the scalar components of A are θ) Ax = A cos ( θ) Ay = A sin ( Scalar Components y Given Ax, Ay, how do you find A, θ ? (-,+ ) (+ ,+ ) x Magnitude: (-,-) (+,-) =+22 AAAxy Angle: Be careful! •Calculator in degree mode? A •Look at the signs of A and A θ = −1 y x y. tan ( ) Does the angle make sense? Ax •Inverse tangent only gives back a result f rom -90° to +90°. How do you get the right quadrant? Adding Vectors by Components +y +y +y C B By C By C C A y Bx θ Ay Ay A A B x +x x x +x Cx +x A+B=C 1. Choose coordinate system and draw a picture. 2. Find scalar components: Ax, Ay, Bx, By 3. Calculate scalar components: Cx =Ax+ Bx and Cy =Ay+ By C =+22 θ = −1 y 4. Find:CCCxy 5. Find: tan ( ) Cx 4 Example N Adding Displacement Vectors: A hikers walks D3 D1 = 6.0 km east on day 1 30° D D2 = 4.0 km north on day 2 T θ D2 D3 = 10.0 km at 30.0° north of west on day 3. W E D Find her total displacement for the 1 Trip. By graphical Methods DT = D1 + D1 + D1 S Example N Analysis x-component y-component D3 30° D1 +6.0 km 0 km DT θ D2 D2 0 km +4.0 km W E D3 -10 cos(30°) +10 sin(30°) D1 = -8.7 km = +5.0 km By graphical Methods DT -2.7 km +9.0 km Dkmkm=−(.)(.)2722 + 90 = 94 . T S − 90. θ =+180ooootan1 ( ) =−=180 73 107 (Needed to add 180° to get result in − 27. the correct quadrant) Vector Description of Motion Average Vector Displacement: ∆ =− rrf ri Average Vector Velocity: ∆r ∆v = ∆t Instantaneous Vector Velocity: A particle moves from P (at ti)to Q(at tf) along the trajectory shown above. ∆r ∆v = lim (A trajectory is a path of motion ∆t→0 ∆t through space) 5 Vector Description of Motion Velocity vectors How does velocity change? shown in red vf vi ∆v ∆ =− vf vvf vi vi Average Vector Acceleration: ∆v ∆a = The velocity vector’s direction is ∆t always tangent to the trajectory and in the direction of the motion. Instantaneous Vector Acceleration: The length of the velocity vector shows ∆v the instantaneous speed of the particle. ∆a = lim ∆t→0 ∆ Longer means faster. t Vector Description of Motion Velocity vectors shown How does velocity change? in red, acceleration vectors in purple vf ai vi ∆v af ∆ =− vf vvf vi vi Average Vector Acceleration: ∆v ∆a = The velocity vector’s direction is ∆t always tangent to the trajectory and in the direction of the motion. Instantaneous Vector Acceleration: The length of the velocity vector shows ∆v the instantaneous speed of the particle. ∆a = lim ∆t→0 ∆ Longer means faster. t Vector Description of Motion As a particle moves along its trajectory, Acceleration vector shows how the velocity vector changes. Velocity vector shows how the position changes Both velocity and acceleration can change in both magnitude and direction (This is a quicktime animation in the screen version ) 6 Projectile Motion Projectile Motion describes the motion of any object thrown into the air at any arbitrary angle. To describe projectile motion mathematically, we assume: 1. Projectile motion is uniform ly accelerated motion. The acceleration vector is directed Quicktime Movie of a juggler vertically downward and has a magnitude of g=9.8 m/s2 . 2. The effects or air resistance are negligible. Analyzing Projectile Motion +y The projectile is launched at θ initial velocity, v0 at angle 0 above the horizontal. v0 (x,y) v Initial components are: θ vv= cosθ 0 +x x00 0 = θ (x0,y0) vvy00sin 0 Subsequent to launch, the x-component of velocity, remains constant; only the y-component changes. x - motion y - motion ==θ = vv=− gt vvxx00 vcos 0constant yy0 xvtv==( cosθ ) t −= −1 2 x000 yy00 vty 2 gt 2 =−2 − vvyy0 2( gyy0 ) The speed at time t is given by: =+22 vvvxy Example: Find components first! ==o vmsmsx0 30cos 30 / 26 / vmsms==30sin 30o / 15 / y0 A ball is thrown off a 25 m high roof at a speed of 30 m/s at an angle of 30° above horizontal. 2 =−⇒=+2 1) What is the maximum height? vgyyyyvgyy0 22()000 / y=+25 m(/)/(./) 15 ms22 19 6 ms = 36 m −= − 2 2) At what time does it land? yy00 vty 05. gt 4..,. 9tt2 −−=⇒=− 15 25 0 t 1 2 sts =+ 4 3 3) How far does it travel? (What is the horizontal range?) −= xx00 vtx xm=+0()(.) 26 4 3 = 110 7 Relative Motion How is a motion described by differerent observers in different reference frames which are in motion with respect to each other? Relative Position Relative Velocity Relative Velocities Consider the motion of a boat in a river. vbr : velocity of boat seen by river vre : velocity of river seen by earth vbe : velocity of boat seen by earth Vector equation: vbe = vbr + vre Sign Convention: vre = - ver Example At what angle do you point the boat to go straight across the river? vbr : speed is 20 m/s, direction ??? vre : current, 10 m/s, due east vbe : speed is ???, due north N Vector equation: vbe = vbr + vre In components: E East: (vbr)x + (vre)x = (vbe)x -20 m/s sin(θ)+10 m/s = 0 ---> θ =30° North: (vbr)y + (vre)y = (vbe)y 30° 20 m/s cos( )+0 m/s = (vbe)y = vbe = 17 m/s 8.
Load more

Recommended publications
  • Linear Algebra I
    Linear Algebra I Martin Otto Winter Term 2013/14 Contents 1 Introduction7 1.1 Motivating Examples.......................7 1.1.1 The two-dimensional real plane.............7 1.1.2 Three-dimensional real space............... 14 1.1.3 Systems of linear equations over Rn ........... 15 1.1.4 Linear spaces over Z2 ................... 21 1.2 Basics, Notation and Conventions................ 27 1.2.1 Sets............................ 27 1.2.2 Functions......................... 29 1.2.3 Relations......................... 34 1.2.4 Summations........................ 36 1.2.5 Propositional logic.................... 36 1.2.6 Some common proof patterns.............. 37 1.3 Algebraic Structures....................... 39 1.3.1 Binary operations on a set................ 39 1.3.2 Groups........................... 40 1.3.3 Rings and fields...................... 42 1.3.4 Aside: isomorphisms of algebraic structures...... 44 2 Vector Spaces 47 2.1 Vector spaces over arbitrary fields................ 47 2.1.1 The axioms........................ 48 2.1.2 Examples old and new.................. 50 2.2 Subspaces............................. 53 2.2.1 Linear subspaces..................... 53 2.2.2 Affine subspaces...................... 56 2.3 Aside: affine and linear spaces.................. 58 2.4 Linear dependence and independence.............. 60 3 4 Linear Algebra I | Martin Otto 2013 2.4.1 Linear combinations and spans............. 60 2.4.2 Linear (in)dependence.................. 62 2.5 Bases and dimension....................... 65 2.5.1 Bases............................ 65 2.5.2 Finite-dimensional vector spaces............. 66 2.5.3 Dimensions of linear and affine subspaces........ 71 2.5.4 Existence of bases..................... 72 2.6 Products, sums and quotients of spaces............. 73 2.6.1 Direct products...................... 73 2.6.2 Direct sums of subspaces................
    [Show full text]
  • Vector Spaces in Physics
    San Francisco State University Department of Physics and Astronomy August 6, 2015 Vector Spaces in Physics Notes for Ph 385: Introduction to Theoretical Physics I R. Bland TABLE OF CONTENTS Chapter I. Vectors A. The displacement vector. B. Vector addition. C. Vector products. 1. The scalar product. 2. The vector product. D. Vectors in terms of components. E. Algebraic properties of vectors. 1. Equality. 2. Vector Addition. 3. Multiplication of a vector by a scalar. 4. The zero vector. 5. The negative of a vector. 6. Subtraction of vectors. 7. Algebraic properties of vector addition. F. Properties of a vector space. G. Metric spaces and the scalar product. 1. The scalar product. 2. Definition of a metric space. H. The vector product. I. Dimensionality of a vector space and linear independence. J. Components in a rotated coordinate system. K. Other vector quantities. Chapter 2. The special symbols ij and ijk, the Einstein summation convention, and some group theory. A. The Kronecker delta symbol, ij B. The Einstein summation convention. C. The Levi-Civita totally antisymmetric tensor. Groups. The permutation group. The Levi-Civita symbol. D. The cross Product. E. The triple scalar product. F. The triple vector product. The epsilon killer. Chapter 3. Linear equations and matrices. A. Linear independence of vectors. B. Definition of a matrix. C. The transpose of a matrix. D. The trace of a matrix. E. Addition of matrices and multiplication of a matrix by a scalar. F. Matrix multiplication. G. Properties of matrix multiplication. H. The unit matrix I. Square matrices as members of a group.
    [Show full text]
  • An Attempt to Intuitively Introduce the Dot, Wedge, Cross, and Geometric Products
    An attempt to intuitively introduce the dot, wedge, cross, and geometric products Peeter Joot March 21, 2008 1 Motivation. Both the NFCM and GAFP books have axiomatic introductions of the gener- alized (vector, blade) dot and wedge products, but there are elements of both that I was unsatisfied with. Perhaps the biggest issue with both is that they aren’t presented in a dumb enough fashion. NFCM presents but does not prove the generalized dot and wedge product operations in terms of symmetric and antisymmetric sums, but it is really the grade operation that is fundamental. You need that to define the dot product of two bivectors for example. GAFP axiomatic presentation is much clearer, but the definition of general- ized wedge product as the totally antisymmetric sum is a bit strange when all the differential forms book give such a different definition. Here I collect some of my notes on how one starts with the geometric prod- uct action on colinear and perpendicular vectors and gets the familiar results for two and three vector products. I may not try to generalize this, but just want to see things presented in a fashion that makes sense to me. 2 Introduction. The aim of this document is to introduce a “new” powerful vector multiplica- tion operation, the geometric product, to a student with some traditional vector algebra background. The geometric product, also called the Clifford product 1, has remained a relatively obscure mathematical subject. This operation actually makes a great deal of vector manipulation simpler than possible with the traditional methods, and provides a way to naturally expresses many geometric concepts.
    [Show full text]
  • Dirac Notation 1 Vectors
    Physics 324, Fall 2001 Dirac Notation 1 Vectors 1.1 Inner product T Recall from linear algebra: we can represent a vector V as a column vector; then V y = (V )∗ is a row vector, and the inner product (another name for dot product) between two vectors is written as AyB = A∗B1 + A∗B2 + : (1) 1 2 ··· In conventional vector notation, the above is just A~∗ B~ . Note that the inner product of a vector with itself is positive definite; we can define the· norm of a vector to be V = pV V; (2) j j y which is a non-negative real number. (In conventional vector notation, this is V~ , which j j is the length of V~ ). 1.2 Basis vectors We can expand a vector in a set of basis vectors e^i , provided the set is complete, which means that the basis vectors span the whole vectorf space.g The basis is called orthonormal if they satisfy e^iye^j = δij (orthonormality); (3) and an orthonormal basis is complete if they satisfy e^ e^y = I (completeness); (4) X i i i where I is the unit matrix. (note that a column vectore ^i times a row vectore ^iy is a square matrix, following the usual definition of matrix multiplication). Assuming we have a complete orthonormal basis, we can write V = IV = e^ e^yV V e^ ;V (^eyV ) : (5) X i i X i i i i i ≡ i ≡ The Vi are complex numbers; we say that Vi are the components of V in the e^i basis.
    [Show full text]
  • 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
    LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set.
    [Show full text]
  • Low-Level Image Processing with the Structure Multivector
    Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.
    [Show full text]
  • Relativistic Inversion, Invariance and Inter-Action
    S S symmetry Article Relativistic Inversion, Invariance and Inter-Action Martin B. van der Mark †,‡ and John G. Williamson *,‡ The Quantum Bicycle Society, 12 Crossburn Terrace, Troon KA1 07HB, Scotland, UK; [email protected] * Correspondence: [email protected] † Formerly of Philips Research, 5656 AE Eindhoven, The Netherlands. ‡ These authors contributed equally to this work. Abstract: A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter. Keywords: invariants; inversion; division; non-division algebra; Dirac algebra; Clifford algebra; geometric algebra; special relativity; photon interaction Citation: van der Mark, M.B.; 1. Introduction Williamson, J.G. Relativistic Inversion, Invariance and Inter-Action.
    [Show full text]
  • 1 Vectors & Tensors
    1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.
    [Show full text]
  • Spacetime Algebra As a Powerful Tool for Electromagnetism
    Spacetime algebra as a powerful tool for electromagnetism Justin Dressela,b, Konstantin Y. Bliokhb,c, Franco Norib,d aDepartment of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USA bCenter for Emergent Matter Science (CEMS), RIKEN, Wako-shi, Saitama, 351-0198, Japan cInterdisciplinary Theoretical Science Research Group (iTHES), RIKEN, Wako-shi, Saitama, 351-0198, Japan dPhysics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA Abstract We present a comprehensive introduction to spacetime algebra that emphasizes its prac- ticality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in stud- ies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials.
    [Show full text]
  • Index Notation
    Index Notation January 10, 2013 One of the hurdles to learning general relativity is the use of vector indices as a calculational tool. While you will eventually learn tensor notation that bypasses some of the index usage, the essential form of calculations often remains the same. Index notation allows us to do more complicated algebraic manipulations than the vector notation that works for simpler problems in Euclidean 3-space. Even there, many vector identities are most easily estab- lished using index notation. When we begin discussing 4-dimensional, curved spaces, our reliance on algebra for understanding what is going on is greatly increased. We cannot make progress without these tools. 1 Three dimensions To begin, we translated some 3-dimensional formulas into index notation. You are familiar with writing boldface letters to stand for vectors. Each such vector may be expanded in a basis. For example, in our usual Cartesian basis, every vector v may be written as a linear combination ^ ^ ^ v = vxi + vyj + vzk We need to make some changes here: 1. Replace the x; y; z labels with a numbers 1; 2; 3: (vx; vy; vz) −! (v1; v2; v3) 2. Write the labels raised instead of lowered. 1 2 3 (v1; v2; v3) −! v ; v ; v 3. Use a lower-case Latin letter to run over the range, i = 1; 2; 3. This means that the single symbol, vi stands for all three components, depending on the value of the index, i. 4. Replace the three unit vectors by an indexed symbol: e^i ^ ^ ^ so that e^1 = i; e^2 = j and e^3 = k With these changes the expansion of a vector v simplifies considerably because we may use a summation: ^ ^ ^ v = vxi + vyj + vzk 1 2 3 = v e^1 + v e^2 + v e^3 3 X i = v e^i i=1 1 We have chosen the index positions, in part, so that inside the sum there is one index up and one down.
    [Show full text]
  • Geometric Intuition
    thesis Geometric intuition Richard Feynman once made a statement represent the consequences of rotations in to the effect that the history of mathematics One day, perhaps, 3-space faithfully. is largely the history of improvements in Clifford’s algebra will It is also completely natural not only to add notation — the progressive invention of or subtract multi-vectors, but to multiply or ever more efficient means for describing be taught routinely to divide them — something not possible with logical relationships and making them students in place of ordinary vectors. The result is always another easier to grasp and manipulate. The Romans multi-vector. In the particular case of a were stymied in their efforts to advance vector analysis. multi-vector that is an ordinary vector V, the mathematics by the clumsiness of Roman inverse turns out to be V/v2, where v2 is the numerals for arithmetic calculations. After to j, that is, eiej = −ejei. Another way to put squared magnitude of V. It’s a vector in the Euclid, geometry stagnated for nearly it is that multiplication between parallel same direction but of reciprocal magnitude. 2,000 years until Descartes invented a new vectors is commutative, whereas it is anti- Write out the components for the notation with his coordinates, which made commutative for orthogonal vectors. product of two vectors U and V, and you it easy to represent points and lines in These rules are enough to define find the resultUV = U•V + ĬU × V, with • space algebraically. the algebra, and it’s then easy to work and × being the usual dot and cross product Feynman himself, of course, introduced out various implications.
    [Show full text]
  • Unit Vector Notation
    Unit Vector Notation 1 Definitions Figure 1 shows the definition of components of a vector V~ . The angles α, β and γ are the angles that the vector makes to the three coordinate axes x, y and z respectively. The dashed lines are perpendiculars drawn from the tip of the vector to the three coordinate axis. The components ~ Vx, Vy and Vz of V are defined to be the segments of the coordinate axes marked out by these perpendiculars as shown. Hence, using the definition of cosine for the three right-angle triangles, we find, Vx = V cos α; Vy = V cos β; Vz = V cos γ; where V (short for jV~ j) is the magnitude of the vector V~ . We also define unit vectors (vectors of magnitude one) along each of the three coordinate axes x, y and z to be ^i, ^j and k^ respectively (figure 2). The following three vectors in the three coordinate directions can now be defined. ~ ^ ~ ^ ~ ^ Vx = Vxi; Vy = Vyj; Vz = Vzk: Using the triangle rule for vector addition twice, this gives, ~ ~ ~ ~ ^ ^ ^ V = Vx + Vy + Vz = Vxi + Vyj + Vzk: This is known as the unit vector notation of a vector. If the vector is restricted to the x-y plane, then γ = 90◦. This makes (figure 3), ◦ Vz = 0; and α + β = 90 : Hence, cos β = cos (90◦ − α) = sin α: Thus the components Vx and Vy can now be written as, Vx = V cos α; Vy = V sin α 1 . y ... ....... ... ... .. ... .. .. .... .. ..... ... .... .... ... ... .. .. ... .. ... ... ..... V . .... .... y . ...... .. ..... ..... ...... .. ... ..... ... .. .... .. ...... .. ~ . ... .. V ...... .... .. β .. ...... .. .. .... .... .. ..... ....... .. .
    [Show full text]