because , we see that this product yields , which links quaternions to the properties of determinants. A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector (strictly speaking a bivector), that is , where a0 is a real number and is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of and as A dual quaternion is usually described as a quaternion with dual numbers as coefficients. A dual number is an ordered pair . Two dual numbers add componentwise and multiply by the rule . Dual numbers are often written in the form , where ε is the dual unit that commutes with i, j, k and has the property . The result is that a dual quaternion can be written as an ordered pair of quaternions . Two dual quaternions add componentwise and multiply by the rule, It is convenient to write a dual quaternion as the sum of a dual scalar and a dual vector, , where and is the dual vector that defines a screw. This notation allows us to write the product of two dual quaternions as Addition The addition of dual quaternions is defined componentwise so that given, and then Multiplication Multiplication of two dual quaternion follows from the multiplication rules for the quaternion units i, j, k and commutative multiplication by the dual unit ε. In particular, given and then Notice that there is no BD term, because the definition of dual numbers requires that . This gives us the multiplication table (note the multiplication order is row times column): Conjugate The conjugate of a dual quaternion is the extension of the conjugate of a quaternion, that is As with quaternions, the conjugate of the product of dual quaternions, , is the product of their conjugates in reverse order, It is useful to introduce the functions Sc(∗) and Vec(∗) that select the scalar and vector parts of a quaternion, or the dual scalar and dual vector parts of a dual quaternion. In particular, if , then This allows the definition of the conjugate of Â as or, The product of a dual quaternion with its conjugate yields This is a dual scalar which is the magnitude squared of the dual quaternion. Dual number conjugate A second type of conjugate