Tensor
A tensor of order n, or simply an n-tensor, is an n-linear function that assigns a scalar to an ordered n-tuple of vectors.
A vector is a 1-tensor. Through the dot product, it linearely assigns a scalar to any vector.
A scalar is a 0-tensor. It assigns a scalar to nothing.
The cross product of two vectors
\( \vec{\mathbf{u}} = \langle u_1, u_2, u_3, u_4 \rangle \)
\( \vec{\mathbf{v}} = \langle v_1, v_2, v_3, v_4 \rangle \)
is a 2-tensor.
Simply by replacing the first and second rows of this determinant
\(
\vec{\mathbf{u}} \times \vec{\mathbf{v}} =
\begin{vmatrix}
\vec{\mathbf{e_1}} & \vec{\mathbf{e_2}} & \vec{\mathbf{e_3}} & \vec{\mathbf{e_4}} \\
\vec{\mathbf{e_1}} & \vec{\mathbf{e_2}} & \vec{\mathbf{e_3}} & \vec{\mathbf{e_4}} \\
u_1 & u_2 & u_3 & u_4 \\
v_1 & v_2 & v_3 & v_4
\end{vmatrix}
\)
with the coordinates of any two vectors, we obtain a \( 4 \times 4 \) determinant of scalars,
thereby bilinearly assigning a scalar to that ordered pair of vectors.