Question

Prove That
$\boxed{ \pink{ \bull\gg \gg \tt |\vec{a} \times \hat{i}|^{2}+|\vec{a} \times \hat{j}|^{2}+|\vec{a} \times \hat{k}|^{2}=2|\vec{a}|^{2} \ll \ll \bull}}$
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Answer 1

$\large\underline{\sf{Solution-}}$

Consider, LHS

${\tt |\vec{a} \times \hat{i}|^{2}+|\vec{a} \times \hat{j}|^{2}+|\vec{a} \times \hat{k}|^{2}}$

Let assume that

$\rm \: \vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$

So, Consider

$\rm \: \vec{a} \times \hat{i}$

$\rm \: = \:(x\hat{i} + y\hat{j} + z\hat{k}) \times \hat{i}$

$\rm \: = \:x(\hat{i} \times \hat{i}) + y(\hat{j} \times \hat{i}) + z(\hat{k} \times \hat{i})$

$\rm \: = \: - y\hat{k} + z\hat{j}$

$\rm\implies \: |\vec{a} \times \hat{i}| = \sqrt{ {y}^{2} + {z}^{2} }$

Hence,

$\rm\implies \: |\vec{a} \times \hat{i}|^{2} = {y}^{2} + {z}^{2} - - - (1)$

Now, Consider

$\rm \: \vec{a} \times \hat{j}$

$\rm \: = \:(x\hat{i} + y\hat{j} + z\hat{k}) \times \hat{j}$

$\rm \: = \:x(\hat{i} \times \hat{j}) + y(\hat{j} \times \hat{j}) + z(\hat{k} \times \hat{j})$

$\rm \: = \:x\hat{k} - z\hat{i}$

$\rm\implies \: |\vec{a} \times \hat{j}| = \sqrt{ {x}^{2} + {z}^{2} }$

Hence,

$\rm\implies \: |\vec{a} \times \hat{j}|^{2} = {x}^{2} + {z}^{2} - - - (2)$

Now, Consider

$\rm \: \vec{a} \times \hat{k}$

$\rm \: = \:(x\hat{i} + y\hat{j} + z\hat{k}) \times \hat{k}$

$\rm \: = \:x(\hat{i} \times \hat{k}) + y(\hat{j} \times \hat{k}) + z(\hat{k} \times \hat{k})$

$\rm \: = \: - x\hat{j} + y\hat{i}$

$\rm\implies \: |\vec{a} \times \hat{k}| = \sqrt{ {x}^{2} + {y}^{2} }$

Hence,

$\rm\implies \: |\vec{a} \times \hat{k}|^{2} = {x}^{2} + {y}^{2} - - - (3)$

Now, on adding equation (1), (2) and (3), we get

${ \tt |\vec{a} \times \hat{i}|^{2}+|\vec{a} \times \hat{j}|^{2}+|\vec{a} \times \hat{k}|^{2}}$

$\rm \: = \: {y}^{2} + {z}^{2} + {x}^{2} + {z}^{2} + {x}^{2} + {y}^{2}$

$\rm \: = \:2( {x}^{2} + {y}^{2} + {z}^{2})$

$\rm \: = \:2 \sqrt{( {x}^{2} + {y}^{2} + {z}^{2}) ^{2}}$

can be rewritten as

$\rm \: = \:2 \: {( \sqrt{ {x}^{2} + {y}^{2} + {z}^{2} } ) \: }^{2}$

$\rm \: = \:2|\vec{a}|^{2}$

Hence,

$\rm\implies \:\boxed{\sf{  \:{\tt |\vec{a} \times \hat{i}|^{2}+|\vec{a} \times \hat{j}|^{2}+|\vec{a} \times \hat{k}|^{2} = 2|\vec{a}|^{2}} \: }}$

$\rule{190pt}{2pt}$

Formulae Used :-

$\boxed{\sf{  \:\rm \: \hat{i} \times \hat{j} = \hat{k} \: }}$

$\boxed{\sf{  \:\rm \: \hat{j} \times \hat{k} = \hat{i} \: }}$

$\boxed{\sf{  \:\rm \: \hat{k} \times \hat{i} = \hat{j} \: }}$

$\boxed{\sf{  \:\rm \: \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\boxed{\sf{  \:\rm \: \vec{a} \times \vec{a} = 0 \: }}$

$\boxed{\sf{  \:\rm \: \vec{a} \times \vec{b} = - \vec{b} \times \vec{a} \: }}$

$\boxed{\sf{  \:\rm \: \vec{a} \times \vec{b} = 0 \: \rm\implies \:\vec{a} \: \parallel \: \vec{b} \: }}$