Here the effect is more complicated, and is best comprehended in the geometrical interpretation to which we proceed after noting three yet more special cases.

Thirdly, we multiply the "real" couple $(a, 0)$ by the imaginary $(0, b)$ and obtain $(a, 0) \times (0, b) = (0, a b) .$

Fourthly, we multiply the two "real" couples $(a, 0)$ and $(a^{'}, 0)$ and obtain $(a, 0) \times (a^{'}, 0) = (a a^{'}, 0) .$

Fifthly, we multiply the two "imaginary couples" $(0, b)$ and $(0, b^{'})$ and obtain $(0, b) \times (0, b^{'}) = (- b b^{'}, 0) .$

We now turn to the geometrical interpretation, beginning first with some special cases.

Take the couples $(1, 3)$ and $(2, 0)$ and consider the equation $(2, 0) \times (1, 3) = (2, 6) .$ 11

In the diagram ([fig.]11) the vector $O P$ represents $(1, 3)$ , and the vector $O N$ represents $(2, 0)$ , and the vector $O Q$ represents $(2, 6)$ . Thus the product $(2, 0) \times (1, 3)$ is found geometrically by taking the length of the vector $O Q$ to be the product of the lengths of the vectors $O P$ and $O N$ , and (in this case) by producing $O P$ to $Q$ to be of the required length. Again, consider the product $(0, 2) \times (1, 3)$ , we have $(0, 2) \times (1, 3) = (- 6, 2) .$

The vector $O N_{1}$ , corresponds to $(0, 2)$ and the vector $O R$ to $(- 6, 2)$ . Thus $O R$ which

represents the new product is at right angles to $O Q$ and of the same length. Notice that we have the same law regulating the length of $O Q$ 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 $O N_{1}$ along the "ordinate" axis $O Y$ , instead of $O N$ along the "abscissa" axis $O X$ , the direction of $O P$ has been turned through a right-angle.

Hitherto in these examples of multiplication we have looked on the vector $O P$ as modified by the vectors $O N$ and $O N_{1}$ . We shall get a clue to the general law for the direction by inverting the way of thought, and by thinking of the vectors $O N$ and $O N_{1}$ as modified by the vector $O P$ . 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 $O N$ (i.e. $O Q$ ) is found by rotating it in the (anti-clockwise) direction of rotation from $O X$ towards $O Y$ through an angle equal to the angle $X O P$ : it is an accident of this particular case that this rotation makes $O Q$ lie along the line $O P$ . Again consider the product of $O N_{1}$ and $O P$ ; the new direction for the enlarged $O N_{1}$ (i.e. $O R$ ) is found by rotating $O N$ in the anti-clockwise direction of rotation through an angle equal to the angle $X O P$ , namely, the angle $N_{1} O R$ is equal to the angle $X O P$ .

The general rule for the geometrical representation of multiplication can now be enunciated thus: 12

The product of the two vectors $O P$ and $O Q$ is a vector $O R$ , whose length is the product of the lengths of $O P$ and $O Q$ and whose direction $O R$ is such that the angle $X O R$ is equal to the sum of the angles $X O P$ and $X O Q$ .

Hence we can conceive the vector $O P$ as making the vector $O Q$ rotate through an angle $X O P$ (i.e. $the angle Q O R = the angle X O P$ ), or the vector $O Q$ as making the vector $O P$ rotate through the angle $X O Q$ (i.e. $the angle P O R = the angle X O Q$ ).

We do not prove this general law, as we