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Question: If \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k},\) \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} ...

If a=3i−j+2k,\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, b=2i+j−k\mathbf{b} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} and c=i−2j+2k,\mathbf{c} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}, then

(a×b)×c(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} is equal to

A

24i+7j−5k24\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}

B

7i−24j+5k7\mathbf{i} - 24\mathbf{j} + 5\mathbf{k}

C

12i+3j−5k12\mathbf{i} + 3\mathbf{j} - 5\mathbf{k}

D

i+j−7k\mathbf{i} + \mathbf{j} - 7\mathbf{k}

Answer

24i+7j−5k24\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}

Explanation

Solution

(a×b)×c=(a.c)b−(b.c)a(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a}.\mathbf{c})\mathbf{b} - (\mathbf{b}.\mathbf{c})\mathbf{a}

=(3+2+4)(2i+j−k)−(2−2−2)(3i−j+2k)= (3 + 2 + 4)(2\mathbf{i} + \mathbf{j} - \mathbf{k}) - (2 - 2 - 2)(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})

=18i+9j−9k+6i−2j+4k=24i+7j−5k= 18\mathbf{i} + 9\mathbf{j} - 9\mathbf{k} + 6\mathbf{i} - 2\mathbf{j} + 4\mathbf{k} = 24\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}.