Home  18.013A  Chapter 3  Section 3.3 


Prove that the dot product is invariant under rotation of coordinates.
Solution:
Because the dot product is linear in each of its arguments, if we prove this
statement for dot products of basis vectors, it will be true for any sum of
them and hence for all vectors.
By symmetry, we need only prove that (i, i) which is 1, is the dot product of
the image of i with itself under rotation of coordinates, and that (i, j), which
is 0, is the dot product of the image of i with that of the image of j. The same
thing will be true, by symmetry with any other choice of basis vectors.
The image of i under rotation in the i, j plane is the form
that i takes in terms
of the rotated i', j' basis vectors, which is i'cos  j'sin. Similarly the
image
of j is j'cos + i'sin.
Taking the dot products of these in terms of the i', j' basis,
we get ,
and cossin
sincos,
which are 1 and 0 as claimed.
