Just as a complex number is a sort of "Two Dimensional Number," so a quaternion is a "Four Dimensional Number."

The quaternions are formed by taking three "things," we call them $i,$ $j,$ and $k,$ and declare them to have the following properties:

• $i^2=-1$
• $j^2=-1$
• $k^2=-1$
• $ijk=-1$
We then form the algebraic closure of these with the real numbers. The result is a set of "things" of the form $a+bi+cj+dj$ where $a,$ $b,$ $c,$ and $d$ are real numbers.

We can add, subtract, and multiply quaternions by using the usual rules of algebra, multiplying out, collecting like terms, and simplifying by using the above relationships. Division of quaternions is more difficult.

The quaternions form a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.

A feature of quaternions is that multiplication of two quaternions is not commutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space[3] or equivalently as the quotient of two vectors.

In the case where the real parts of quaternions $q$ and $q'$ are both zero, we can consider the quaternions to be analogous to three dimensional vectors.

In this case we can write $q~=~xi+yj+zk$ and $q'~=~x'i+y'j+z'k$ as quaternions. The quaternion product $qq'$ then works out to be

$-(xx'+yy'+zz')~+~(yz'-zy')i~+~(zx'-xz')j~+~(xy'-yx')k$

The real part is the negative of the dot product (scalar product) of the three dimensional vectors $(x,y,z)$ and $(x',y',z'),$ and the vector part is the vector cross product of $(x,y,z)$ and $(x',y',z').$

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