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**Hint:**In order to find a vector, first of all consider a general vector. Let us consider the given vector to be ${{\overrightarrow r = x \hat i + y \hat j}}$. We know that if two vectors are mutually perpendicular to each other then their dot product is zero. Now, find the dot product of both the vectors. Relate the obtained equation and the given equation to find the components of the vector. And the components of the vector are found by substitution method.

**Complete step by step solution:**

Let us consider that the required vector is ${{\overrightarrow r = x \hat i + y \hat j}}$

According to the question, ${{x \hat i + y \hat j}}$ is perpendicular to ${{\overrightarrow A = \hat i + 2 \hat j}}$.

When one vector is perpendicular to the other vector then the dot product of both the vectors is zero.

Thus, the dot product of vector ${{\overrightarrow r = x \hat i + y \hat j}}$ and vector ${{\overrightarrow A = \hat i + 2 \hat j}}$ must be zero.

Now, finding the dot product of both the vectors

$

\Rightarrow {{\overrightarrow r}}{{. \overrightarrow A = (x \hat i + y \hat j)}}{{.(\hat i + 2 \hat j)}} \\

\Rightarrow {{x + 2 y = 0}} \\

\Rightarrow {{x = - 2 y}}...{{(i)}} $

Given that the magnitude of vector r is ${{3}}\sqrt {{5}} $

So, ${{|r| = }}\sqrt {{{{x}}^{{2}}}{{ + }}{{{y}}^{{2}}}} {{ = 3}}\sqrt 5 $

Squaring both sides, we get

$

\Rightarrow {\left( {\sqrt {{{{x}}^{{2}}}{{ + }}{{{y}}^{{2}}}} } \right)^2}{{ = }}{\left( {{{3}}\sqrt 5 } \right)^2} \\

\Rightarrow {{{x}}^{{2}}}{{ + }}{{{y}}^{{2}}}{{ }} = {{ }}45$

Substituting the value of x from (i), we get

$

\Rightarrow {{{( - 2 y)}}^{{2}}}{{ + }}{{{y}}^{{2}}}{{ = 45}} \\

\Rightarrow {{5 }}{{{y}}^2}{{ }} = {{ }}45 \\

\therefore {{y = 3}} $

Now substituting this value of y in equation (i), we get

$

\Rightarrow {{x = - 2 y = - 2 (3)}} \\

\Rightarrow {{x = - 6}}$

Hence, the required vector becomes

$\Rightarrow {{\overrightarrow r = 6 \hat i - 3 \hat j}}$

**Therefore, option (B) is the correct choice.**

**Note:**In a two-dimensional coordinate system, any vector can be broken into x - component and y - component. Let us consider the general vector, ${{\overrightarrow r = x \hat i + y \hat j + \hat z k}}$. But in the question, we have assumed the general vector to be ${{\overrightarrow r = x \hat i + y \hat j}}$. This is because of the fact that the other vector which is provided in the question stem i.e. ${{\hat i + 2 \hat j}}$ does not involve a vector of z - component. Each part of the two-dimensional vector is also known as a component. The components of a given vector depicts the influence of that vector in a given direction.

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