If $\theta $ is the angle between $\bar{a}$ and $\bar{b}$, then $\displaystyle \dfrac{\left | \bar{a} \times \bar{b}\right |}{\bar{a}\cdot \bar{b}}$ equals

Using the definitions of the scalar and vector products, the scalar product a dot b equals the magnitude of a times the magnitude of b times cos theta. The magnitude of the cross product a cross b equals the magnitude of a times the magnitude of b times sin theta. Substituting these into the given expression yields the magnitude of a times the magnitude of b times sin theta divided by the magnitude of a times the magnitude of b times cos theta. This simplifies to sin theta divided by cos theta, which equals tan theta.

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