Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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 each other in either of the following ways.

      (1.9)

1.3 Vector Products

      1.3.1 Dot Product

      The dot product (a.k.a. scalar product) of two vectors and is denoted and defined as follows:

(1.10)

      In Eq. (1.10), θ_pq is defined as the angle between the vectors and . It is denoted as

      (1.11)

      Without any significant loss of generality, the range of θ_pq may be defined so that 0 ≤ θ_pq ≤ π. According to this range definition, it happens that

(1.12)

      Besides, cosθ_pq is not sensitive to the sense of θ_pq anyway. Therefore, the order of the vectors in the dot product is immaterial. That is,

      (1.13)

      If , i.e. if is perpendicular (or orthogonal or normal) to so that θ_pq = π/2, then .

      If