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

If a=i+j−k,b=i−j+k,c=i−j−k\mathbf{a} = \mathbf{i} + \mathbf{j} - \mathbf{k},\mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k},\mathbf{c} = \mathbf{i} - \mathbf{j} - \mathbf{k}, then a×(b×c)\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) is

A

i – j + k

B

2i – 2j

C

3i – j + k

D

2i + 2j – k

Answer

2i – 2j

Explanation

Solution

∵a×(b×c)=(a.c)b−(a.b)c\because a \times (b \times c) = (a.c)b - (a.b)c

∵a.c=(i+j−k).(i−j−k)=1−1+1=1\because a.c = (i + j - k).(i - j - k) = 1 - 1 + 1 = 1

a.b=(i+j−k).(i−j+k)=1−1−1=−1a.b = (i + j - k).(i - j + k) = 1 - 1 - 1 = - 1

∴a×(b×c)=(1)b−(−1)ca \times (b \times c) = (1)b - ( - 1)\mathbf{c}

=b+c=i−j+k+i−j−k=2i−2j= \mathbf{b} + \mathbf{c} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \mathbf{i} - \mathbf{j} - \mathbf{k} = 2\mathbf{i} - 2\mathbf{j}.