The vector space $ \mathbb{V}$ is introduced in Simple tensor algrbra I, several rules are provided in it. To reduce the abstract, we represent the vectors $ \boldsymbol{x}, \boldsymbol{y}$ with the scalar coefficients $ \{{x_1},{x_2},{x_3} \}$ and $ \{{y_1},{y_2},{y_3} \}$ in $ \mathbb{E}^3$, respectively. The composition of two vectors $ \boldsymbol{x}, \boldsymbol{y} \in\mathbb{E}^3$ is expanded by,

**Dot product ($ \cdot $):**

$${\boldsymbol{x}} \cdot {\boldsymbol{y}} = {x_i}{{\boldsymbol{g}}^i} \cdot {y_j}{{\boldsymbol{g}}^j} = {x_i}{y_j}{{\boldsymbol{g}}^i} \cdot {{\boldsymbol{g}}^j} = {x_i}{y_j}{g^{ij}} = {x^i}{y_i} = \left[ {\begin{array}{*{20}{c}}

{{x_1}}&{{x_2}}&{{x_3}}

\end{array}} \right]\left[ {\begin{array}{*{20}{c}}

{{y_1}}\\

{{y_2}}\\

{{y_3}}

\end{array}} \right] = \sum\limits_{i = 1}^3 {{x_i}} {y_i}$$

The *dot product* is also called *scalar product* or *inner product* under Cartesian coordinates and it provides the Euclidean magnitudes of these two vectors and the cosine of the angle between them.

**Cross product ($ \times$):**

$${\boldsymbol{x}} \times {\boldsymbol{y}} = {x_i}{{\boldsymbol{g}}^i} \times {y_j}{{\boldsymbol{g}}^j} = {x_i}{y_j}{{\boldsymbol{g}}^i} \times {{\boldsymbol{g}}^j} = {x_i}{y_j}{\varepsilon ^{ijk}}g{{\boldsymbol{g}}_k} = \left| {\begin{array}{*{20}{c}}

{{x_1}}&{{x_2}}&{{x_3}}\\

{{y_1}}&{{y_2}}&{{y_3}}\\

{{{\boldsymbol{e}}_1}}&{{{\boldsymbol{e}}_2}}&{{{\boldsymbol{e}}_3}}

\end{array}} \right|$$

The *cross product* is also called *vector product*, and it represents the area of a parallelogram with the sides of these two vectors. Now, a new composition for two vectors $ \boldsymbol{x}, \boldsymbol{y} \in\mathbb{E}^3$ is introduced here.

This is not a tutorial, please refer to the books for tensor algebra.

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