Let $\overrightarrow a$, $\overrightarrow b$, $\overrightarrow c$ be three non-coplanar vectors such that $\overrightarrow a$ $\times$ $\overrightarrow b$ = 4$\overrightarrow c$, $\overrightarrow b$ $\times$ $\overrightarrow c$ = 9$\overrightarrow a$ and $\overrightarrow c$ $\times$ $\overrightarrow a$ = $\alpha$$\overrightarrow b$, $\alpha$ > 0. If $$\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| + \left| {\overrightarrow c } \right| = {1 \over {36}}$$, then $\alpha$ is equal to __________.

<p>Given,</p> <p>$\overrightarrow a \times \overrightarrow b = 4\,.\,\overrightarrow c$ ..... (i)</p> <p>$\overrightarrow b \times \overrightarrow c = 9\,.\,\overrightarrow a$ ..... (ii)</p> <p>$\overrightarrow c \times \overrightarrow a = \alpha \,.\,\overrightarrow b$ .... (iii)</p> <p>Taking dot products with $\overrightarrow c ,\overrightarrow a ,\overrightarrow b$ we get</p> <p>$$\overrightarrow a \,.\,\overrightarrow b = \overrightarrow b \,.\,\overrightarrow c = \overrightarrow c \,.\,\overrightarrow a = 0$$</p> <p>Hence,</p> <p>(i) $$ \Rightarrow |\overrightarrow a |\,.\,|\overrightarrow b | = 4\,.\,|\overrightarrow c |$$ ..... (iv)</p> <p>(ii) $$ \Rightarrow |\overrightarrow b |\,.\,|\overrightarrow c | = 9\,.\,|\overrightarrow a |$$ ..... (v)</p> <p>(iii) $$ \Rightarrow |\overrightarrow c |\,.\,|\overrightarrow a | = \alpha \,.\,|\overrightarrow b |$$ .... (vi)</p> <p>Multiplying (iv), (v) and (vi)</p> <p>$$ \Rightarrow |\overrightarrow a |\,.\,|\overrightarrow b |\,.\,|\overrightarrow c | = 36\alpha $$ ..... (vii)</p> <p>Dividing (vii) by (iv) $$ \Rightarrow |\overrightarrow c {|^2} = 9\alpha \Rightarrow |\overrightarrow c | = 3\sqrt \alpha $$ ..... (viii)</p> <p>Dividing (vii) by (v) $$ \Rightarrow |\overrightarrow a {|^2} = 4\alpha \Rightarrow |\overrightarrow a | = 2\sqrt \alpha $$</p> <p>Dividing (viii) by (vi) $\Rightarrow |\overrightarrow b {|^2} = 36 \Rightarrow |\overrightarrow b | = 6$</p> <p>Now, as given, $$3\sqrt \alpha + 2\sqrt \alpha + 6 = {1 \over {36}} \Rightarrow \sqrt \alpha = {{ - 43} \over {36}}$$</p>