Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

, i.e. if |q| = |p| and θ_pq = θ_pp = 0, then . Hence, the magnitude of a vector can also be expressed as

      (1.14)

      1.3.2 Cross Product

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

(1.15)

      In Eq. (1.15), as defined before, θ_pq is the angle measured from to . As for , it is defined as a unit vector, which is perpendicular to the plane formed by the vectors and .

If and are skew (nonparallel) vectors, then is formed by imagining that and are translated toward each other until they are connected tail‐to‐tail. If and are parallel (but not coincident) vectors, then happens to be the plane that contains them. However, if and are coincident vectors, then cannot be formed as a definite plane, i.e. it can be any plane that contains them.

      The sense of is defined conventionally by the right‐hand rule. This rule is based on the right hand in such a way that assumes the orientation of the thumb (directed from root to tip) while the fingers are oriented from to .

      Since, by definition, and , the following equations can be written for the vectors involved in the cross product.

      (1.16)

      (1.17)

      If sin θ_pq = 0, i.e. if with θ_pq = 0 or with θ_pq = π, then

      (1.18)

      If the order of and is reversed, Eq. (1.15) becomes

      (1.19)

      According to Eq. (1.12), θ_qp = θ_pq. However, according to the right‐hand rule,

      (1.20)

      Therefore, . This verifies the well‐known characteristic feature of the cross product that its outcome changes sign when the order of its multiplicands is reversed. That is,

      (1.21)

1.4 Reference Frames

In the three‐dimensional Euclidean space, a reference frame is defined as an entity that consists of an origin and three distinct noncoplanar axes emanating from the origin. The origin is a specified point and the axes have specified orientations. More specifically, the axes of a reference frame are called its coordinate axes. For the sake of verbal brevity, a reference frame may sometimes be called simply a frame. A reference frame, such as the one shown in Figure 1.1, may be denoted in one of the following ways, which convey different amounts of information about its specific features.

(1.22)

Figure 1.1 A reference frame.

      In Eq. (1.22), A is the origin of . The origin of may also be denoted as O_a. The coordinate axes of are oriented so that each of them is aligned with one member of the following set of three vectors, which is denoted as and defined as the basis vector triad of