A vector is a coordinate invariant quantity, uniquely defined by a magnitude and a direction in space, and that obeys the parallelogram law by addition.
The magnitude of a vector \( \mathbf{a} \) is denoted: $$ \begin{equation} \mathbf{|a|} \equiv a \tag{8.3} \end{equation} $$
A vector \( \mathbf{a} \) may be decomposed into scalar vector components or for short components, \( a_i \), parallel to the coordinate axes, i.e., parallel to the base vectors \( \mathbf{e}_i \), and then be presented in various, equivalent ways: $$ \begin{equation} \mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 \equiv a_i \mathbf{e}_i \equiv (a_1, a_2, a_3) \equiv [a_i] \equiv \left [ \begin{array}{c} a_1\\ a_2\\ a_3 \end{array} \right ] \tag{8.4} \end{equation} $$
Addition and subtraction of vectors are defined according to the geometric parallelogram law [2] (see section 2.2), but the geometrical definition is transformed to the component form given by: $$ \begin{equation} \mathbf{a + b = c} \Leftrightarrow a_i + b_i = c_i \tag{8.5} \end{equation} $$
The scalar product (or dot product) of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined by: $$ \begin{equation} \mathbf{a \cdot b} = \mathbf{|a||b|} \cos (\mathbf{a,b}) \tag{8.6} \end{equation} $$ where $ (\mathbf{a,b})$ is the angle between the two vectors. The operation is commutative and distributive. For orthogonal base vectors in a coordinate system \( Ox \) we get: $$ \begin{equation} \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij} \tag{8.7} \end{equation} $$ where \( \delta_{ij} \) is the Kronecker delta defined as: $$ \begin{equation} \delta_{ij} = \left \{ \begin{array}{ll} 1& \text{when } i=j \\ 0 & \text{when } i\not=j \end{array} \right . \tag{8.8} \end{equation} $$
From these fundamental definitions and properties one may show[2] (see section 2.2): $$ \begin{equation} \mathbf{a \cdot b} = a_i b_i \tag{8.9} \end{equation} $$
The *vector product*or cross product is defined by: $$ \begin{equation} \mathbf{a \times b = c} \equiv \mathbf{|a||b|} \sin(\mathbf{(a,b) \, e} \tag{8.10} \end{equation} $$ where \( \mathbf{(a,b)} \) is the smallest angle between the two vectors \( \mathbf{a} \) and \( \mathbf{b} \), and \( \mathbf{e} \) is a unit vector orthogonal to the plane defined by \( \mathbf{a} \) and \( \mathbf{b} \). By introducing the permutation symbol: $$ \begin{equation} e_{ijk} = \left \{ \begin{array}{rl} 0& \text{when two or three indices are equal} \\ 1 & \text{when the indices are cyclic permutations of 123} \\ -1 &\text{when the indices are cyclic permutations of 321} \end{array} \right . \tag{8.11} \end{equation} $$ one may also conveniently express the vector product by: $$ \begin{equation} \mathbf{a \times b = c} \qquad \Leftrightarrow \qquad e_{ijk} \, a_i b_j = c_k \tag{8.12} \end{equation} $$