An introduction to mathematics, by A. N. Whitehead.

IMAGINARY NUMBERS 107
represents the new product is at right angles
to OQ and of the same length. Notice that
we have the same law regulating the length
of OQ as in the previous case, namely, that
its length is the product of the lengths of
the two vectors which are multiplied together; but now that we have ON1 along the
"ordinate" axis OY, instead of ON along
the "abscissa" axis OX, the direction of OP
has been turned through a right-angle.
Hitherto in these examples of multiplication
we have looked on the vector OP as modified
by the vectors ON and ON1. We shall get
a clue to the general law for the direction by
inverting the way of thought, and by thinking of the vectors ON and ON1 as modified by
the vector OP. The law for the length remains unaffected; the resultant length is the
length of the product of the two vectors.
The new direction for the enlarged ON (i.e.
OQ) is found by rotating it in the (anti-clockwise) direction of rotation from OX towards
OY through an angle equal to the angle POX:
it is an accident of this particular case that
this rotation makes OQ lie along the line OP.
Again consider the product of ON1 and OP;
the new direction for the enlarged ON1 (i.e.
OR) is found by rotating ON in the anticlockwise direction of rotation through an
angle equal to the angle POX, namely, the
angle N1OR is equal to the angle POX.