Chapter 4: Programming in Matlab - College of the Redwoods

396 Chapter 4 Programming in Matlab
ij = k = −ji, jk = i = −kj, ki = j = −ik (4.11)
Students of vector calculus will recognize the familiar relationships amongst i, j,
and k, if they think of i, j, and k as unit vectors along the x, y, and z axes in the
usual 3-space orientation used in vector calculus.
Of course, the relationships in (4.11) demonstrate that multiplication of the
quaternions is not commutative. Changing the order of the factors changes the
product. In general, if u and v are quaternions, it is not the case that uv equals
vu.
Sir William was able to demonstrate that multiplication was an associative operation
((uv)w = u(vw)) and that multiplication is distributive with respect to
addition (u(v + w) = uv + uw). With associativity and the distributive law in
hand, a straightforward (albeit messy) calculation shows that the product of the
quaternions u = u 1 + u 2 i + u 3 j + u 4 k and v = v 1 + v 2 i + v 3 j + v 4 k is
uv = (u 1 v 1 − u 2 v 2 − u 3 v 3 − u 4 v 4 ) + (u 1 v 2 + u 2 v 1 + u 3 v 4 − u 4 v 3 )i
+ (u 1 v 3 + u 3 v 1 + u 4 v 2 − u 2 v 4 )j + (u 1 v 4 + u 4 v 1 + u 2 v 3 − u 3 v 2 )k.
(4.12)
When you studied the complex numbers, before making the definition of division,
you paused to develop the complex conjugate. In a similar manner, if u = u 1 +
u 2 i + u 3 j + u 4 k, then the conjugate of this quaternion is the quaternion
u = u 1 + u 2 i + u 3 j + u 4 k = u 1 − u 2 i − u 3 j − u 4 k. (4.13)
Note that the conjugate of a quaternion if very similar to the conjugate of a
complex number. That is, if z is the complex number z = a + bi, then the
conjugate of z is z = a − bi. Geometrically, the conjugate of the complex number
z is the reflection of z across the real axis, as shown in Figure 4.11.
Imag
z = a + bi
Real
z = a − bi
Figure 4.11. The conjugate is a reflection
across the real axis.