First slide

Scalars and Vectors

Question

$If c = \hat{i} \times (a \times \hat{i}) + \hat{j} \times (a \times \hat{j}) + \hat{k} \times (a \times \hat{k}) and c = n a . The$ $value of n is _______$

Difficult

Solution

$\begin{array}{l} Let a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k} \\ ∴ c = \hat{i} \times (a \times \hat{i}) + \hat{j} \times (a \times \hat{j}) + \hat{k} (a \times \hat{k}) \\ = \hat{i} \times (- a_{2} \hat{k} + a_{3} \hat{j}) + \hat{j} \times (a_{1} \hat{k} - a_{3} \hat{i}) + \hat{k} \times (- a_{1} \hat{j} + a_{2} \hat{i}) \\ = a_{2} \hat{j} + a_{3} \hat{k} + a_{1} \hat{i} + a_{3} \hat{k} + a_{1} \hat{i} + a_{2} \hat{j} \\ = 2 (a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}) = 2 a \\ ∴ n = 2 \end{array}$

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