In mathematics, a quaternion biradial[1](Art.93) is the quotient (or product $\mathbf{b}/\mathbf{a}$, $\mathbf{b}\mathbf{a}$, $\mathbf{a}\mathbf{b}$, $\mathbf{a}/\mathbf{b}$) of two pure quaternion vectors $\mathbf{a}$ and $\mathbf{b}$, sometimes called rays.
Consider the quaternion biradial $b=\mathbf{b}/\mathbf{a}$ (read “b by a”). The biradial $b$ is an operator that turns $\mathbf{a}$ into $\mathbf{b}$ as an operator on $\mathbf{a}$: $\mathbf{b}=b\mathbf{a}=\mathbf{b}\mathbf{a}^{-1} \mathbf{a}=\mathbf{b}$. The biradial turning operation is a composition of a rotation and a scaling that rotates $\mathbf{a}$ into the line of $\mathbf{b}$, followed with scaling by $| \mathbf{b} | / | \mathbf{a} |$. If $| \mathbf{a} | = | \mathbf{b} |$ then $| b | =1$ and $b$ acts as only a rotation operator (a.k.a., a versor, rotor, or 2D spinor) in the $\mathbf{a}\mathbf{b}$-plane through the angle $\theta$ from $\mathbf{a}$ to $\mathbf{b}$.