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Step 1: Express the vector in component form
Let $ \overrightarrow{a} = x\,\hat{i} + y\,\hat{j} + z\,\hat{k} $. Here, $x$, $y$, and $z$ are the scalar components of $ \overrightarrow{a} $ along the $x$, $y$, and $z$ axes respectively.
Step 2: Compute the cross products
$ \overrightarrow{a} \times \hat{i} = (x\,\hat{i} + y\,\hat{j} + z\,\hat{k}) \times \hat{i} $.
Since $ \hat{i} \times \hat{i} = \overrightarrow{0} $, $ \hat{j} \times \hat{i} = -\hat{k} $, and $ \hat{k} \times \hat{i} = \hat{j}$,
we get $ \overrightarrow{a} \times \hat{i} = z\,\hat{j} - y\,\hat{k} $.
$ \overrightarrow{a} \times \hat{j} = (x\,\hat{i} + y\,\hat{j} + z\,\hat{k}) \times \hat{j} $.
Using $ \hat{j} \times \hat{j} = \overrightarrow{0} $, $ \hat{k} \times \hat{j} = -\hat{i} $, and $ \hat{i} \times \hat{j} = \hat{k}$,
we get $ \overrightarrow{a} \times \hat{j} = x\,\hat{k} - z\,\hat{i} $,
but rearranging in the standard order ($\hat{i}, \hat{j}, \hat{k}$), we write it as
$ \overrightarrow{a} \times \hat{j} = -z\,\hat{i} + x\,\hat{k} $.
$ \overrightarrow{a} \times \hat{k} = (x\,\hat{i} + y\,\hat{j} + z\,\hat{k}) \times \hat{k} $.
Using $ \hat{k} \times \hat{k} = \overrightarrow{0} $, $ \hat{i} \times \hat{k} = -\hat{j} $, and $ \hat{j} \times \hat{k} = \hat{i}$,
we get $ \overrightarrow{a} \times \hat{k} = y\,\hat{i} - x\,\hat{j} $.
Step 3: Calculate the squared magnitudes
$ (\overrightarrow{a} \times \hat{i})^2 = (z\,\hat{j} - y\,\hat{k})^2 = z^2 + (-y)^2 = y^2 + z^2. $
$ (\overrightarrow{a} \times \hat{j})^2 = (-z\,\hat{i} + x\,\hat{k})^2 = (-z)^2 + x^2 = x^2 + z^2. $
$ (\overrightarrow{a} \times \hat{k})^2 = (y\,\hat{i} - x\,\hat{j})^2 = y^2 + (-x)^2 = x^2 + y^2. $
Step 4: Sum the squared magnitudes
Add all three results:
$ (\overrightarrow{a} \times \hat{i})^2 + (\overrightarrow{a} \times \hat{j})^2 + (\overrightarrow{a} \times \hat{k})^2
= (y^2 + z^2) + (x^2 + z^2) + (x^2 + y^2). $
Combine like terms:
$ = (x^2 + x^2) + (y^2 + y^2) + (z^2 + z^2) = 2(x^2 + y^2 + z^2). $
Since $x^2 + y^2 + z^2 = \overrightarrow{a}^2$, we get
$ (\overrightarrow{a} \times \hat{i})^2 + (\overrightarrow{a} \times \hat{j})^2 + (\overrightarrow{a} \times \hat{k})^2 = 2\,\overrightarrow{a}^2.
Step 5: State the final result
The required value is $ 2 \overrightarrow{a}^2 $.
