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$\mathrm {A}$ and $\mathrm {B}$ and expressed in natural measure^[1]. This involves that the same rotation may be represented in many different ways by two vectors in the plane $V$.

For the rotation $\mathrm {R}$ we shall also use the symbol $[\mathrm {A\cdot B]}$.

By the vector product $[\mathrm {A\cdot B\cdot C]}$ of three vectors $\mathrm {A,B,C}$ at a point of the field-figure and not lying in one plane we shall understand a vector $\mathrm {D}$ the direction of which is conjugate with each of the three vectors (and therefore with the three-dimensional extension $\mathrm {A,B,C}$), the direction of $\mathrm {D}$ corresponding to those of $\mathrm {A,B}$ and $\mathrm {C}$ in a way presently to be indicated, while the magnitude of $\mathrm {D}$, expressed in natural measure, is equal to that of the parallelepiped described on $\mathrm {A}$, $\mathrm {B}$ and $\mathrm {C}$ and expressed in the same measure. This definition involves that the value is ascribed to the vector product of three vectors lying in one and the same plane.

A further statement about the direction of $\mathrm {D}$ is necessary because two opposite directions are conjugate with $\mathrm {A,B,C}$. For one set of three directions $\mathrm {A_{0},B_{0},C_{0}}$ we shall choose arbitrarily which of its two conjugate directions will be said to correspond to it. If this is the direction $\mathrm {D} _{0}$, then the direction $\mathrm {D}$ corresponding to $\mathrm {A,B,C}$ will be determined by the rule that $\mathrm {D} _{0}$, passes into $\mathrm {D}$ by a gradual passage of the first three vectors from $\mathrm {A_{0},B_{0},C_{0}}$ into $\mathrm {A,B,C}$, this latter passage being effected in such a way that during the change the vectors never come to lie in one plane.

The vector product $[\mathrm {A\cdot B\cdot C]}$ takes the opposite direction when one of the vectors is reversed as well as when two of them are interchanged. We must therefore always attend to the order of the symbols in $[\mathrm {A\cdot B\cdot C]}$.

The vector product possesses the distributive property with respect to each of the three vectors, so that e.g. if $\mathrm {A} _{1}$ and $\mathrm {A} _{2}$ are vectors,

$\left[\left(\mathrm {A} _{1}+\mathrm {A} _{2}\right)\cdot \mathrm {B\cdot C} \right]=\mathrm {\left[A_{1}\cdot B\cdot C\right]+\left[A_{2}\cdot B\cdot C\right]}$

From this we can infer that $[\mathrm {A\cdot B\cdot C]}$ depends only on $\mathrm {C}$ and the rotation $\mathrm {R}$ determined by $\mathrm {A}$ and $\mathrm {B}$. For this reason we write for the vector product also $[\mathrm {R\cdot C]}$; in calculating it we are free to replace the rotation $\mathrm {R}$ by any two vectors by means of which it can be represented.

If $\mathrm {R}$, $\mathrm {R} _{1}$ and $\mathrm {R} _{2}$ are rotations in the same plane, such that the value and direction of $\mathrm {R}$ are found by adding $\mathrm {R} _{1}$ and $\mathrm {R} _{2}$ algebraically, we have, in virtue of the distributive property

$[\mathrm {R_{1}\cdot C]} +[\mathrm {R_{2}\cdot C]} =[\mathrm {R\cdot C]}$

1. If, according to circumstances, different signs arc given to $\mathrm {R}$, the angle whose sine occurs in the formula for the area of a parallelogram must be understood to be positive in one case and negative in the other.