Now, some imagination is required. Let’s return to the 3-dimensional Cartesian system. At any
point P, we can specify three local axes and three local planes determined by these axes. In
accordance with strict definitions, the axes must be mutually perpendicular and, by extension, so
must the planes. Now, choose three unit vectors at P such that each vector is tangent to one of
the axes. Such a triple is usually designated (i, j, k). Any vector V at P can then be written
V = αi + βj + γk
where α, β, and γ are the usual x, y, and z scalar components of the vector.
Now suppose that we had chosen unit vectors perpendicular to each of the planes rather than
tangent to each of the coordinate axes. Let’s do so and call the resulting triple (i*, j*, k*). Again,
any vector V at P can be written
V = α*i* + β*j* + γ*k*
where α*, β*, and γ* are the scalar components of the vector referred to the i*, j*, k* triple.
There is nothing surprising in what we have just done, and our representation is satisfactory
provided we ensure that
αi + βj + γk = α*i* + β*j* + γ*k*.
But, you might argue that what we have done is trivial since it is apparent from geometry that the
two unit vector triples comprise the same set; i.e., that
i = i*
j = j*
k = k*.
Still, we used two distinct approaches to defining a unit vector triple at P. Should we expect
these approaches to produce so tidy a result in all cases? The answer is very definitely “NO”!
To understand why the answer is “NO,” let’s modify our Cartesian system so that the axes are no
longer mutually orthogonal – for example, so that they meet at 60°. In this case, the origin lies at
a vertex of a tetrahedron, and the axes lie along three of the edges. (Such coordinate systems are actually used in engineering and crystallography and are called triangular coordinate systems.) It should be intuitive that (i, j, k) and (i*, j*, k*) are now two different sets of unit vectors.
Specifically, i and i* now meet at an angle of 60°, as do j and j*, and k and k*. Thus, while they
are all unit vectors, they specify different sets of directions, and the choice of which set to use in
a given calculation must be a matter of expediency.
