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The Cross Product

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We saw in the previous section on dot products that the dot product takes two vectors and produces a scalar, making it an example of a scalar product. In this section, we will introduce a vector product, a multiplication rule that takes two vectors and produces a new vector. We will find that this new operation, the cross product, is only valid for our 3-dimensional vectors, and cannot be defined in the 2- dimensional case. The reasons for this will become clear when we discuss the kinds of properties we wish the cross product to have.

Rotational Invariance

One important feature of the dot product which we didn't mention in the previous section is its invariance under rotations. In other words, if we take a pair of vectors in the plane and rotate them both by the same angle (imagine, for instance, that the vectors are sitting on a record, and rotate the record), their dot product will remain the same. Consider the length of a single vector (which is given by the dot product): if the vector gets rotated about the origin by some angle, its length will not change--even though its direction can change quite dramatically! Similarly, from the geometric formula for the dot product, we see that the result depends only on the lengths of the two vectors and the angle between them. None of these quantities changes when we rotate the two vectors together, so neither can their dot product. This is what we mean when we say that the dot product is invariant under rotations.

Rotational invariance ends up being a very important property in physics. Imagine writing down vector equations to describe some physical situation taking place on a table. Now rotate the table (or keep the table fixed, and rotate yourself by some angle around the table). You haven't really changed anything about the physics on the table by simply turning everything by some fixed angle. Because of this, you should expect your equations to retain their form. This means that if these equations involve products of vectors, these products better be rotationally invariant. The dot product has already passed this test, as we noted above. We now want to require the same of the cross product.

Making the requirement of rotational invariance more stringent for the cross product, we need the cross product of two vectors to yield another vector. Consider, for instance, two 3-dimensional vectors u and v in a plane (two non-parallel vectors always define a plane, in the same way that two lines do. If we rotate this plane, the vectors will change direction, but we don't want the cross product w = u×v to change at all. However, if w has any non-zero components in the plane of u and v , those components will necessarily change under rotation (they get rotated just like everything else). The only vectors that won't change at all under a rotation of the u - v plane are those vectors that are perpendicular to the plane. Hence, the cross product of two vectors u and v must give a new vector which is perpendicular to both u and v .

This simple observation actually goes a long way towards constraining our options for how we can define the cross product. For instance, we can see immediately that it is not possible to define a cross product for two- dimensional vectors, since there is no direction perpendicular to the plane of two-dimensional the vectors! (We'd need a third dimension for that).

Now that we know the direction in which the cross product of two vectors points, the magnitude of the resulting vector remains to be specified. If I take the cross product of two vectors in the x - y plane, I now know that the resulting vector should point purely in the z -direction. But should it point upwards (i.e. lie along the positive z -axis) or should it point downwards? How long should it be?

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