If $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}; \vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}, \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k} and [3 \vec{a} + \vec{b} 3 \vec{b} + \vec{c} 3 \vec{c} + \vec{a}]$ $= λ |\begin{array}{ccc} \vec{a} \cdot \hat{i} & \vec{a} \cdot \hat{j} & \vec{a} \cdot \hat{k} \\ \vec{b} \cdot \hat{i} & \vec{b} \cdot \hat{j} & \vec{b} \cdot \hat{k} \\ \vec{c} \cdot \hat{i} & \vec{c} \cdot \hat{j} & \vec{c} \cdot \hat{k} \end{array}|$ then the value of $\frac{λ}{4}$

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Solution

$\begin{array}{l} \vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k} \\ \vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k} \\ \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k} \\ \begin{array}{l} \end{array} \end{array}$
$L.H.S. = [\begin{array}{lll} 3 \vec{a} + \vec{b} 3 \vec{b} + \vec{c} 3 \vec{c} + \vec{a} \end{array}]$
$\begin{array}{l} = [3 \vec{a} 3 \vec{b} 3 \vec{c}] + [\vec{b} \vec{c} \vec{a}] \\ = 3^{3} [\vec{a} \vec{b} \vec{c}] + [\vec{a} \vec{b} \vec{c}] \\ = 28 [\vec{a} \vec{b} \vec{c}] \end{array}$

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