Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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a vector equation can be written without indicating any reference frame, whereas the selected reference frame must be indicated for the corresponding matrix equation.

1.1 General Features of Notation

      This section gives general information about the special notation that is used throughout the book. This notation is convenient because it can be used not only in printed texts but also in handwritten work. It also has the desirable feature that it can distinguish column matrices from vectors, which are actually different mathematical objects. The main features of the notation are explained below.

       A scalar is denoted by a plain letter such as s.

       A vector is denoted by a letter with an overhead arrow such as .

       A column matrix is denoted by a letter with an overhead bar such as .

       A square or a rectangular matrix is denoted by a capital letter with an overhead circumflex (a.k.a. hat) such as .

       A skew symmetric matrix is denoted by a letter with an overhead tilde such as .

       The transpose of a matrix is denoted by a superscript t such as and .

1.2 Vectors

      1.2.1 Definition and Description of a Vector

A vector is defined as an entity that can be described by a magnitude and a direction. In this book, it is assumed that all the vectors belong to the three‐dimensional Euclidean space.

      The magnitude of a vector is denoted as shown below.

      (1.1)

      A unit vector such as is defined so that its magnitude is unity. That is,

      (1.2)

      A vector can be expressed as follows by means of a unit vector , which is introduced to indicate the direction of .

(1.3)

      In Eq. (1.3), v is defined as the scalar value of with respect to .

      Note that the magnitude of a vector is the absolute value of its scalar value. That is,

      (1.4)

      Note also that the scalar value v can be positive, negative, or zero, but the magnitude can only be positive or zero.

      The sign variability of the scalar value is demonstrated in the following equation.

(1.5)

      According to Eq. (1.5), the scalar values of the same vector with respect to and are v and v^′ = − v, respectively.

      1.2.2 Equality of Vectors

      Two vectors and are defined to be equal, i.e. , if they satisfy the following equations simultaneously, in which .

(1.6)

(1.7)

      In Eqs. (1.6) and (1.7), σ is a sign variable such that

      (1.8)

      In Eq. (1.7), the notation indicates the direction of the vector . Equation (1.7) implies the following two situations for the unit vectors and .

      If σ = + 1, and are codirectional, i.e. either coincident or parallel with the same direction.

      If σ = − 1, and are opposite, i.e. either coincident or parallel with opposite directions.

      1.2.3 Opposite Vectors

Two vectors and are defined to be opposite, i.e. , if they are related to

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