If $\overrightarrow{a}.\overrightarrow{b}=1,\overrightarrow{b}.\overrightarrow{c}=2\overrightarrow{c}.\overrightarrow{a}=3,$ then the value of $\left[\overrightarrow{a}\times \left(\overrightarrow{b}\times \overrightarrow{c}\right),\overrightarrow{b}\times \left(\overrightarrow{c}\times \overrightarrow{a}\right),\overrightarrow{c}\times \left(\overrightarrow{b}\times \overrightarrow{a}\right)\right]is:$

<p>&nbsp;<math><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>×</mo><mrow><mo> (</mo><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>−</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mi>u</mi><mo>→</mo></mover></mrow></math></p><p><math><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>×</mo><mrow><mo> (</mo><mrow><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>a</mi><mo>→</mo></mover></mrow><mo>)</mo></mrow><mo>=</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>−</mo><mn>2</mn><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mi>v</mi><mo>→</mo></mover></mrow></math></p><p><math><mrow><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>×</mo><mrow><mo> (</mo><mrow><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>a</mi><mo>→</mo></mover></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>−</mo><mn>2</mn><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mi>w</mi><mo>→</mo></mover></mrow></math></p><p><math><mrow><mo>∴</mo><mover accent="true"><mi>u</mi><mo>→</mo></mover><mo>+</mo><mover accent="true"><mi>v</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mi>w</mi><mo>→</mo></mover></mrow></math></p><p>so vectors&nbsp;</p><p><math><mrow><mover accent="true"><mi>u</mi><mo>→</mo></mover><mo>, </mo><mover accent="true"><mi>v</mi><mo>→</mo></mover><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>w</mi><mo>→</mo></mover></mrow></math></p><p>are coplanar, hence their Scalar triple product will be zero.</p>

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