As in the case of the scalar product, one of the most important

properties of the vector product is its distribtUivity, i.e. for any three

vectors A, B, C,

VA(B +C) = VAB + VAC. (20)

This capital property can be proved in a variety of ways. First
of all, by an immediate geometrical construction of both the
right- and the left-hand member of (19), — ^which will be left as
an exercise for the reader. (It will be enough if the reader con-
structs it for the simplest case of coplanar A, B, C.*) Another,
comparatively simple proof, based upon (19), is this:

Let us write

VA(B+CJ-VAB-VAC=X.
Then our problem is reduced to proving that X vanishes. Now,

all the three addends being perpendicular to A, so is their sum

X, i.e.
XA»o.
Again, XB»BVA(B + 0-BVAC

= (B-hOVBA-CVBA, by {19),

=BVBA + CVBA - CVBA »o,

and similarly XC»o. Thus, the vector X either vanishes or is
normal to each of the three vectors A, B, C. Now, if these are
not coplanar, the latter case is excluded, so that X»0. Thus,
for non-coplanar A, B, C the distributive property (20) is already
proved. And if A, B, C happen to be coplanar, add to C, for
instance, a fourth vector D inclined to the plane of A, B, C. Then
the new vectors A, B, C+D will not be coplanar, and

VA(C+D)+VB(C+D)=V(A+B)(C+D),

* For the case of non-coplanar A, B, C is more easily dealt with by the
following analytical method.
