State and explain the characteristics of the vector product of two vectors.

(N/A) $(1)$ The vector product is anticommutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$. It does not follow the commutative law.
$(2)$ The vector product is distributive over addition: $\vec{a} \times (\vec{b} + \vec{c}) = (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c})$.
$(3)$ For two parallel or anti-parallel vectors,the vector product is zero: $\vec{a} \times \vec{a} = |a||a| \sin(0^{\circ}) \hat{n} = \vec{0}$.
$(4)$ For two perpendicular vectors,the magnitude of the vector product is maximum: $|\vec{a} \times \vec{b}| = |a||b| \sin(90^{\circ}) = |a||b|$.
$(5)$ The vector product of two vectors is a pseudovector (axial vector). Under reflection,the sign of the vector product does not change because both vectors change sign: $(-\vec{a}) \times (-\vec{b}) = \vec{a} \times \vec{b}$.
$(6)$ For unit vectors in a Cartesian coordinate system: $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0}$ and $\hat{i} \times \hat{j} = \hat{k}$,$\hat{j} \times \hat{k} = \hat{i}$,$\hat{k} \times \hat{i} = \hat{j}$.