The dot product (a.k.a. scalar product) of two vectors and is denoted and defined as follows:
(1.10)
In Eq. (1.10), θpq is defined as the angle between the vectors and . It is denoted as
(1.11)
Without any significant loss of generality, the range of θpq may be defined so that 0 ≤ θpq ≤ π. According to this range definition, it happens that
(1.12)
Besides, cosθpq is not sensitive to the sense of θpq 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 θpq = π/2, then .
If , i.e. if |q| = |p| and θpq = θpp = 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, θpq 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 θpq = 0, i.e. if with θpq = 0 or with θpq = π, then
following ways.
