Let a,b,c be three non zero vectors such that no two of them are collinear and (a*b)*c=1/3|b||c|a. If Q is the angle betwwen b and c then what is the value of sinQ ?

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Given}, \\ \left(\mathrm{a}\times \mathrm{b}\right)\times \mathrm{c}=-\mathrm{a}\left(\mathrm{b}.\mathrm{c}\right)+\mathrm{b}\left(\mathrm{a}.\mathrm{c}\right)=-\mathrm{a}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{cosQ}+\mathrm{b}\left|\mathrm{a}\right|\left|\mathrm{c}\right|\mathrm{cosP} \\ \mathrm{given}, \\ \left(\mathrm{a}\times \mathrm{b}\right)\times \mathrm{c}=\frac{1}{3}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a} \\ -\mathrm{a}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{cosQ}+\mathrm{b}\left|\mathrm{a}\right|\left|\mathrm{c}\right|\mathrm{cosP}=\frac{1}{3}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a} \\ \mathrm{comparing} \mathrm{both} \mathrm{sides} \mathrm{we} \mathrm{get} \\ \mathrm{cosP}=0 \\ \mathrm{hence}, \\ -\mathrm{a}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{cosQ}=\frac{1}{3}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a} \\ \mathrm{or} \mathrm{cosQ}=-\frac{1}{3} \\ \mathbf{sinQ}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{cos}}^{\mathbf{2}}\mathbf{Q}}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}\frac{\mathbf{1}}{\mathbf{9}}}\mathbf{=}\frac{\sqrt{\mathbf{8}}}{\mathbf{3}} \end{gathered}$$

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