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Question: If \(\mathbf{u} = \mathbf{i} \times (\mathbf{a} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{a} \...

If u=i×(a×i)+j×(a×j)+k×(a×k),\mathbf{u} = \mathbf{i} \times (\mathbf{a} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{a} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{a} \times \mathbf{k}), then

A

u=0\mathbf{u} = 0

B

u=i+j+k\mathbf{u} = \mathbf{i} + \mathbf{j} + \mathbf{k}

C

u=2a\mathbf{u} = 2\mathbf{a}

D

u=a\mathbf{u} = \mathbf{a}

Answer

u=2a\mathbf{u} = 2\mathbf{a}

Explanation

Solution

Let a=xi+yj+zk\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}

i×(a×i)+j×(a×j)+k×(a×k)\mathbf{i} \times (\mathbf{a} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{a} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{a} \times \mathbf{k})

=(i.i)a−i(a.i)+(j.j)a−j(a.j)+(k.k)a−k(a.k)\mathbf{= (i.i)a}\mathbf{-}\mathbf{i(a.i) + (j.j)a}\mathbf{-}\mathbf{j(a.j) + (k.k)a}\mathbf{-}\mathbf{k(a.k)}

=3a−a=2a.\mathbf{= 3a}\mathbf{-}\mathbf{a = 2a.}