2.6 Vector products and changes of coordinates in E3 101coordinates of 0' are xo, yo, zo, so that
(2.66) 00' = xoi + yoj + zok,
and that the unit vectors i, j, k and 1', j', k' of the two coordinate systems
are related by the fundamental formulas (2.65). It is now surprisingly
easy to solve our problem. Let P be a point in E3 having unprimed
coordinates x, y, z and primed coordinates x', y', z', so that
(2.661) OP = xi + yj + zk, 0'P = x'i' + y'j' + z'k'.
Putting the formula OP = 00' + 0'P in the form OP - 00' = 0'P
then gives
(2.662) (x - xo)i + (y- yo)j + (z -zo)k = x'i' + y'j' + z'k'.
An expression giving the right side in terms of i, j, k is obtained by
multiplying the members of the first three equations in (2.65) by x', y', a',
respectively, and adding the results. The coefficients of i, j, and k turn
out to be, respectively, the right members of the equations
x - xo = alix' + ally' + a3iz'
Y - yo = a12x' + a22y' + a32z'
z - zo = aiax' + a23Y' + a33z'.These equations therefore result from equating the coefficients of i, j,
and k in (2.662). Transposing gives the formulasx = xo + aiax' + aziy' +a31%'
(2.67) y = yo + ai2x' + a22Y' + a32z
z = zo + aiax' + any' +a33z'which express the unprimed coordinates of a point in terms of the primed
coordinates of the point. A very similar procedure gives the formulasx' = xo + aiax + any + aiax
(2.671) Y' = y' + a2ix + a22Y + a23z
Z' = zp + a31x + a32Y + a33zwhich express the primed coordinates of a point in terms of the unprimed
coordinates of the point. The formulas (2.67) and (2.671) are known
as the formulas for changes of coordinates. The formulas (2.67) are
often used to convert an equation involving coordinates x, y, z into a
new equation involving new coordinates x', y', z'. As can be expected,
it is sometimes far from easy to so determine the numbers xo, yo, zo
and a,, in (2.67) that the new equation will have the simplest possible
form. For the present we do not need to know much about these
matters, but we should know that there are situations in which one