\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.

This can be written in terms of the four standard basis vectors (e1, e2, e3, e4), given by

\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix},

so the general vector a is

\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.

Vectors add, subtract and scale as in three dimensions. The dot product also generalizes to four dimensions, like so:

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.

It can be used to calculate the norm or length of a vector,

\left| \mathbf{a} \right| = \sqrt{\mathbf{a} \cdot \mathbf{a} } = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},

and calculate or define the angle between two vectors as

\theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.

The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows

\mathbf{a} \wedge \mathbf{b} = (a_1b_2 – a_2b_1)\mathbf{e}_{12} + (a_1b_3 – a_3b_1)\mathbf{e}_{13} + (a_1b_4 – a_4b_1)\mathbf{e}_{14} + (a_2b_3 – a_3b_2)\mathbf{e}_{23} + (a_2b_4 – a_4b_2)\mathbf{e}_{24} + (a_3b_4 – a_4b_3)\mathbf{e}_{34}.

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12, e13, e14, e23, e24, e34). They can be used to generate rotations in four dimensions.

thats the math behind it! =]