CHAPTER III.

PRODUCTS OF TWO VECTORS.

107. Notation. So far we have treated only of the Addition of Vectors and some immediate applications. We have now to further develope the Algebra of Vectors.

It has been found advantageous to consider two distinct products of vectors, called the Vector-Product and the Scalar-Product respectively, the one being a Vector and the other a Scalar. As these products are essentially different, it is necessary to distinguish them by a difference in notation. It seems most convenient to do this by aid of brackets. If a and B are any two vectors, then

[aß] shall mean the Vector Product of a and ß,

(a) shall mean the Scalar Product of a and B.

108. Hamilton denoted these products by placing the letters V and S before them, but his scalar product is the negative of ours, so that

[aß]= Vaß and (aß)=– Saß.

109. In § 15 the product of two lengths a and b was defined as the area of a rectangle obtained by placing the lengths perpendicular to each other and then moving one along the other. The name product was justified by the fact that this geometrical product followed the laws of the algebraical product.

In order to get a product of two vectors we shall repeat this process, but in doing so we must take account of the

fact that each vector has a definite direction which we cannot alter without altering the vector.

Let then two vectors a and ẞ be given (being vectors we may consider them as starting from a common origin 0) and suppose ẞ moved along a.

A parallelogram (see fig.) OACB is thus swept out, and it seems appropriate to take its area as a product of two vectors. It will be found that this is an extremely useful conception, with many applications.

Let us study this product a little more carefully. First of all, each vector has a sense and this obliges us to take the area in a definite sense. This is determined (compare § 22) by giving a sense to the boundary. We write this product [as] so that the first factor a is the one supposed fixed whilst the second factor B moves along it, and we take the sense of the parallelogram in the sense which the first factor gives to the boundary.

It will now be seen, on drawing the figures, that the area changes its sense if the sense of either factor be changed, and that it has the original sense again if the sense of both factors be changed. This corresponds to a change in sign of the factors of an algebraical product.

If however we keep ẞ fixed and move a along it, we get the same parallelogram, but now the boundary has the sense of B, and this is opposite to the sense of a. This shews that the products [aß] and [Ba] are not the same. They are equal in magnitude but opposite in sense and we write

[aẞ] = - [Ba].

This product therefore changes sign if the factors be interchanged, and therefore does not agree in all respects

with the laws of the algebraical product:-The law of commutation does not hold for the Vector Product of two Vectors.

Whatever origin O be chosen for the vectors, the parallelogram obtained by the above process has always the same area and sense, and is in one of a number of parallel planes. The aspect of the parallelogram being determined by the direction perpendicular to the parallel planes (see SS 23 and 24) the vector product of two vectors has a definite magnitude and a definite direction and sense connected with it, and is therefore a vector.

a and B being vectors whose magnitudes are lengths, the vector product [a] is a vector whose magnitude is

an area.

Corresponding to our line and area vectors, Maxwell used the names force and flux vectors.

If a and ẞ denote other physical vector quantities, then the magnitude of [aß] is of the same kind as the product of the magnitudes of a and B and has the same dimensions. In particular, if a and B are orts, then [aß]

is an ort.

In each case the magnitude of the product is supposed set off to scale, in the proper sense, along the normal to the plane determined by a and B, the product [aß] is then represented by a line or geometrical vector.

We are now in a position to give a formal definition of the product as follows:

110. The Vector-Product [aß] of two vectors a and B is a vector perpendicular to both, its length represents to scale the area of the parallelogram generated by moving the second vector along the first, and the area is taken in the sense of the first vector.

The sense of [aß] is connected with the sense of the area boundary as the forward motion is to the turning motion of a right-handed screw.

111. In the figure, if a and ẞ are in the plane of the paper, [a] sticks upwards. If a points E. and 6 N., then [a] is towards the zenith, while if a points W., then [a] points vertically downwards.

Let y=[aẞ], then a, ß, and y are three vectors and form what is called a right-handed system.

A good rule to remember the relationship between the vectors forming a right-handed system is the following:

α

Fig. 76.

imagine oneself standing in one vector, the positive sense being from feet to head, and to look along the second, then the third will be to the left. Stand in a, look along B, y is to the left; stand in B, look along y, and a is to the left; stand in y, look along a, then B is to the left. Thus the vectors may be taken in the cyclical order

aßy, Bya, or yaß. (Fig. 76.)

If any other order be taken the system is called left-handed.

If a, B, y are mutually perpendicular, then the three form what is called a right system.

If 1, 2, 3 be three mutually perpendicular orts forming a right system, then

and

[462] = 13

(The student should draw the figures corresponding to these cases for himself.)

112. It should be noticed that it is the magnitude of the area, not its shape, which determines the magnitude of [aß].

Two vector products are equal if the corresponding areas are of equal size in parallel planes and have the

same sense.

For instance, the magnitude of [aß] might have been defined as the area of the rectangle formed by a and the projection* B2 of ẞ on a perpendicular to a. This rectangle being in the plane of a and B, if we give its boundary the sense of a, we have an equivalent definition to that in § 110 and [aß] = [aß2].

Fig. 77.

113. The Scalar Product. The last way of looking at the vector product leads us naturally to consider a rectangle formed by a and the projection of ẞ on a.

* Projection means orthogonal projection unless otherwise specified.