Answer

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**Hint:**First find the value of a in terms of x, y, z by applying dot product on $\left| \overrightarrow{a} \right|$. After that find the values of $\left( \overrightarrow{a}\times \hat{i} \right)$, $\left( \overrightarrow{a}\times \hat{j} \right)$ and $\left( \overrightarrow{a}\times \hat{k} \right)$ by applying cross product rule. Then square each term and add them. Now take commonly from them and substitute the values derived from the $\left| \overrightarrow{a} \right|$.

**Complete step by step answer:**

Given: $\left| \overrightarrow{a} \right|=a$

Let the vector be $\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}$.

Since $\left| \overrightarrow{a} \right|=a$. Then,

$\Rightarrow \sqrt{{{\left( x\hat{i} \right)}^{2}}+{{\left( y\hat{j} \right)}^{2}}+{{\left( z\hat{k} \right)}^{2}}}=a$

As we know the dot product of any unit vector with any other is zero, $\hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{k}\cdot \hat{i}=0$. So,

$\Rightarrow \sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}={{a}^{2}}$

Square both sides of the equation,

$\Rightarrow {{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{a}^{2}}$ …………….….. (1)

Now,

$\Rightarrow \overrightarrow{a}\times \hat{i}=\left( x\hat{i}+y\hat{j}+z\hat{k} \right)\times \hat{i}$

Multiply the terms on the right side,

$\Rightarrow \overrightarrow{a}\times \hat{i}=x\hat{i}\times \hat{i}+y\hat{j}\times \hat{i}+z\hat{k}\times \hat{i}$

Since the cross product must be perpendicular to the two-unit vectors, it must be equal to the other unit vector or the opposite of that unit vector. The cross product of any unit vector with itself is zero, $\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=0$.

$\Rightarrow \overrightarrow{a}\times \hat{i}=x\left( 0 \right)+y\left( -\hat{k} \right)+z\left( {\hat{j}} \right)$

Multiply the terms,

$\Rightarrow \overrightarrow{a}\times \hat{i}=-y\hat{k}+z\hat{j}$

Similarly,

$\Rightarrow \overrightarrow{a}\times \hat{j}=x\hat{k}-z\hat{i}$

$\Rightarrow \overrightarrow{a}\times \hat{k}=-x\hat{j}+y\hat{i}$

Substitute the values,

$\Rightarrow {{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}={{\left( -y\hat{k}+z\hat{j} \right)}^{2}}+{{\left( x\hat{k}-z\hat{i} \right)}^{2}}+{{\left( -x\hat{j}+y\hat{i} \right)}^{2}}$

As we know the dot product of any unit vector with any other is zero, $\hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{k}\cdot \hat{i}=0$. So,

$\Rightarrow {{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}={{\left( -y\hat{k} \right)}^{2}}+{{\left( z\hat{j} \right)}^{2}}+{{\left( x\hat{k} \right)}^{2}}+{{\left( -z\hat{i} \right)}^{2}}+{{\left( -x\hat{j} \right)}^{2}}+{{\left( y\hat{i} \right)}^{2}}$

Square the terms,

$\Rightarrow {{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}={{y}^{2}}+{{z}^{2}}+{{x}^{2}}+{{z}^{2}}+{{x}^{2}}+{{y}^{2}}$

Add the lime terms,

$\Rightarrow {{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}=2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}$

Take 2 commons from the right side,

$\Rightarrow {{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}=2\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}} \right)$

Substitute the value from equation (1),

$\Rightarrow {{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}=2{{a}^{2}}$

**Hence, the value of ${{\left( \overrightarrow{a}\times \hat{i} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{j} \right)}^{2}}+{{\left( \overrightarrow{a}\times \hat{k} \right)}^{2}}$ is $2{{a}^{2}}$.**

**Note:**

The vector can be multiplied by,

Dot product:- The dot product of two vectors is the magnitude of one time the projection of the second onto the first.

Since a vector has no projection perpendicular to itself, the dot product of any unit vector with any other is zero.

$\hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=1$

Cross product:- The cross product of two vectors is the area of the parallelogram between them.

The cross product of any unit vector with itself is zero.

$\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=0$

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