Answer
58.5k+ views
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.
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.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
One mole of which of the following has the highest class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Formula for number of images formed by two plane mirrors class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the pH of 001 M solution of HCl a 1 b 10 c class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If a wire of resistance R is stretched to double of class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)