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## The vector product

We saw earlier, in Sect. 3.10, that it is possible to combine two vectors multiplicatively, by means of a scalar product, to form a scalar. Recall that the scalar product of two vectors and is defined
 (319)

where is the angle subtended between the directions of and .

Is it also possible to combine two vector multiplicatively to form a third (non-coplanar) vector? It turns out that this goal can be achieved via the use of the so-called vector product. By definition, the vector product, , of two vectors and is of magnitude

 (320)

The direction of is mutually perpendicular to and , in the sense given by the right-hand grip rule when vector is rotated onto vector (the direction of rotation being such that the angle of rotation is less than ). See Fig. 70. In coordinate form,
 (321)

There are a number of fairly obvious consequences of the above definition. Firstly, if vector is parallel to vector , so that we can write , then the vector product has zero magnitude. The easiest way of seeing this is to note that if and are parallel then the angle subtended between them is zero, hence the magnitude of the vector product, , must also be zero (since ). Secondly, the order of multiplication matters. Thus, is not equivalent to . In fact, as can be seen from Eq. (321),

 (322)

In other words, has the same magnitude as , but points in diagrammatically the opposite direction.

Now that we have defined the vector product of two vectors, let us find a use for this concept. Figure 71 shows a rigid body rotating with angular velocity . For the sake of simplicity, the axis of rotation, which runs parallel to , is assumed to pass through the origin of our coordinate system. Point , whose position vector is , represents a general point inside the body. What is the velocity of rotation at point ? Well, the magnitude of this velocity is simply

 (323)

where is the perpendicular distance of point from the axis of rotation, and is the angle subtended between the directions of and . The direction of the velocity is into the page. Another way of saying this, is that the direction of the velocity is mutually perpendicular to the directions of and , in the sense indicated by the right-hand grip rule when is rotated onto (through an angle less than ). It follows that we can write
 (324)

Note, incidentally, that the direction of the angular velocity vector indicates the orientation of the axis of rotation--however, nothing actually moves in this direction; in fact, all of the motion is perpendicular to the direction of .

Next: Centre of mass Up: Rotational motion Previous: Is rotation a vector?
Richard Fitzpatrick 2006-02-02