Unit Vector

5 Key excerpts on "Unit Vector"

• 

  eBook - ePub

  Applied Engineering Analysis

  • Tai-Ran Hsu(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Figure 3.3 Vector in rectangular and cylindrical coordinate systems. (a) A vector in three-dimensional space. (b) Rectangular Unit Vectors.

  Figure 3.3 b illustrates another way to represent a vector, in this case in three-dimensional space; for instance, the position vector A with its initial point coinciding with the origin of the coordinate system and its terminal point situated at point P at a position with coordinates (x,y,z) in a rectangular coordinate system. This position vector may be expressed in the following way:

  3.3

  where i, j, and k are the “Unit Vectors” along the x-, y-, and z-coordinate directions, respectively, as shown Figure 3.3 b. Unit Vectors are used to designate the direction of scalar quantities. For example, in Equation 3.3 , the term yj indicates the vector quantity with a magnitude of y but in the direction of the vector j along the y-coordinate. Each of these three Unit Vectors in Figure 3.3 b has a magnitude of 1.0.

  The magnitude of vector A can be obtained by using the Pythagorean rule:

  3.4

  3.2.1 Position Vectors

  A position vector such as the vector A of a point located at (x1 , y1 ) illustrated in Figure 3.3 a. The position of this point may be represented by a vector r in Figure 3.4 a. This vector often is known as the location vector of the point P in space, or the radius vector that represents the position of point P in a space defined by a coordinate system with an arbitrary reference origin O in Figure 3.4 b. The position vector is used in describing the motion of “particles” or “rigid bodies” in a planar two-dimensional space or in three-dimensional space.

  Figure 3.4

  Position vectors. (a) In the two-dimensional plane. (b) In three-dimensional space.

  The position vector r in Figure 3.4 b can be decomposed into three components as r = rx + ry + rz , with rx , ry , and rz being the components of the position vector r along the x-, y-, and z-coordinates, respectively. These components may be expressed using the Unit Vectors i, j, and k as defined in Figure 3.3 b, with rx = x1 i, ry = y1 j, and rz = z1 k

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  eBook - ePub

  Essentials of Physics Series

  An Introduction

  • Derek Raine(Author)
  • 2018(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  4

  VECTORS

  The physical world (as we perceive it) exists in three spatial dimensions. Physical quantities such as velocities – which have both magnitude and direction – need three components for their specification. We call these vectors . (In fact, even more than this, a chemical system, or an economic model may have, say, 20 components; such a system can often be represented conveniently by a vector in the appropriate number of dimensions (20 in this example): the algebra is the same.) You are going to learn to do algebra with vectors (add and multiply them) in a geometrically meaningful way.

  The first part of the chapter deals with the algebra and the second with the geometrical picture. You can think of this in two ways: either that the algebra enables you to do complicated geometrical things rather mechanically (by following the algebraic rules), or that the geometry enables you to get a picture of the algebra. In either case you have to learn to switch smoothly between the two methods, and this requires practice. Your reward will be command of a mathematical tool that pervades physics.

  FIGURE 4.1: (a) The vectors and . (b) The parallelogram law of vector addition.

  4.1.BASIC PROPERTIES

  Definition: A vector is an object with a magnitude and a direction.

  Therefore any vector can be represented by a line segment with the specified length (or magnitude) and direction. Note that this picture provides a representation of any physical quantity, for example a force, that can be described by a magnitude and direction in space.It does not say that a vector is an arrow, only that they behave geometrically in the same way. (Bodies are accelerated by forces, not by arrows on a page.) This is important: many physical quantities are not represented by spatial arrows (velocities and forces in relativity theory, for example).

  A line segment with end-points A and B can be written . The line segment is a vector of the same magnitude but opposite direction (indicated by the direction of the arrowhead in Figure 4.1

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  eBook - ePub

  Electromagnetics and Transmission Lines

  Essentials for Electrical Engineering

  • Robert Alan Strangeway, Steven Sean Holland, James Elwood Richie(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  We begin with how vectors can be used to describe location and displacement. What is a vector ? It is a quantity with magnitude (scalar) and direction. What are examples of vectors? Velocity, force, acceleration, and so forth. How is direction in three‐dimensions expressed? Start with the following vector definition: A position vector locates a position in space with respect to the origin, that is, it is a vector that starts at the origin and ends at the point of the designated position in space (notation r is also used in some textbooks and literature). Example 1.1.1 How is the position vector to point P (2,3) expressed? Strategy Solution What is the magnitude? Try the Pythagorean Theorem: (see footnote 1) Note: no bar over a vector variable means it is the magnitude of the vector, that is, What is the direction of ? Express it in terms of known directions. Figure 1.1 Cartesian (or rectangular) Unit Vectors. Let’s interrupt Example 1.1.1 with some discussion on vector direction. How are known directions expressed? By a Unit Vector : a vector with unit (one) magnitude. Thus, when a Unit Vector is multiplied by a scalar, it creates a vector in the desired direction with a magnitude equal to the scalar. Unit Vector notation : (or a x), where: (or a) indicates the vector is a Unit Vector (a without a bar or not bold is something else, often a dimension or a radius), and subscript x indicates the direction of the Unit Vector. See Figure 1.1 for the three Unit Vectors in the Cartesian coordinate system. Example 1.1.1 resumed: direction of How does one form a Unit Vector in an arbitrary direction? Again, use known directions: “Start at 0, go 2 units in the x direction and 3 units in the y direction to reach P” Thus, is the position vector from 0 to P, as shown in Figure 1.2. How is expressed as (magnitude of) (direction of)? The, as before. The is

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  u” stands for “Unit Vector”).

  To define 2-dimensional Unit Vectors, create the following expressions for each, leave the insertion point in each expression and click the New Definition button.

  When you click the New Definition button, the Interpret Subscript dialog box will appear. Click the Part of the name radio button, then click OK. Now you can define a vector in terms of the Unit Vectors.

  Once you’ve defined your vector, you can Evaluate it and SNB returns the vector in matrix form.

  You can use Simplify or Evaluate to calculate its magnitude.

  Example 3.12

  May the net force be with you

  Write the net force vector from the Tug-of-war example in unit-vector notation.

  Solution.

  The x-component of the net force is 142.64 N and the y-component is 81.915 N. In unit-vector notation, the net force looks like this:

  You can still use Simplify to calculate its magnitude.

  You can create a Unit Vector in the direction of any particular vector by dividing that vector by its magnitude. The Unit Vector pointing in the direction of vector is

  (3.8)

  The Unit Vector is parallel to , it is dimensionless, and has magnitude 1.

  Example 3.13

  One step at a time

  Find the Unit Vector pointing in the direction of the position vector {12 m 5 m}.

  Solution.

  Create an expression for the vector (in either matrix or unit-vector notation) divided by its magnitude, Simplify it, and apply Evaluate Numerically to the result.

  We can verify that this is a Unit Vector by calculating its magnitude with Evaluate .

  Multiplying Vectors

  Multiplication involving vectors shows up in many physics problems. You can multiply a vector by a scalar to produce a vector, and you can multiply two vectors to produce a scalar, a vector, or a second-rank tensor.

  In our discussion of vector multiplication, we will use 3-dimensional vectors. As in the 2-d case, once we define the 3-dimensional Unit Vectors, we can write and manipulate vectors in unit-vector notation.

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  eBook - ePub

  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  surface normal at a given point on an object is a vector that is perpendicular to the surface at that location. However, since the concept of perpendicular is related only to the direction of a vector and not its magnitude, in most cases you will find that Unit Vectors are used for normals instead of a vector of arbitrary length. When this book refers to a vector as a “normal,” it means “a Unit Vector perpendicular to something else.” This is common usage, but be warned that the word “normal” primarily means “perpendicular” and not “unit length.” Since it is so common for normals to be Unit Vectors, we will take care to call out any situation where a “normal” vector does not have unit length.

  In summary, a “normalized” vector always has unit length, but a “normal” vector is a vector that is perpendicular to something and by convention usually has unit length.

  2.9.1   Official Linear Algebra Rules

  For any nonzero vector v , we can compute a Unit Vector that points in the same direction as v . This process is known as normalizing the vector. In this book we use a common notation of putting a hat symbol over Unit Vectors; for example,

  v ^

  (pronounced “v hat”). To normalize a vector, we divide the vector by its magnitude:

  Normalizing a vector

  v ˆ

  =

  v

  ‖ v ‖

  for any nonzero vector  v .

  For example, to normalize the 2D vector [12, −5],

  [ 1 2 - 5 ]

  ‖ [ 1 2 - 5 ] ‖

  =

  [ 12 - 5 ]

  1

  2 2

  +

  5 2

  =

  [ 1 2 - 5 ]

  1 6 9

  =

  [ 1 2 - 5 ]

  1 3

  = [

  12 13

  − 5

  13

  ]

  ≈ [ 0 . 9 2 3   - 0 . 3 8 5 ] .

  The zero vector cannot be normalized. Mathematically, this is not allowed because it would result in division by zero. Geometrically, it makes sense because the zero vector does not define a direction—if we normalized the zero vector, in what direction should the resulting vector point?

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