Answer
64.8k+ views
Hint: We can define the work done as the product of force applied on a body and displacement produced by it. Now, put the values of force and displacement and evaluate the dot product of force and displacement. Also assume the initial position of the particle to be at origin.
Complete step by step answer:
Let the force applied on a body be $\vec F$, displacement produced by the body be $\vec s$ and the work done by the body be $\vec W$.
According to the question, it is given that –
$\implies \vec F = \left( {3\hat i - 2\hat j + \hat k} \right)N$
$\implies \vec s = \left( {2\hat i - 4\hat j + c\hat k} \right)m$ and
$\implies \vec W = 16J$
Work done is the transfer of energy for the displacement of an object using the application of force. Work has only magnitude and no direction therefore, it is the scalar quantity. The S.I unit of work is Joule. It transfers energy from one place to another.
Now, we know that the work done can be defined as the product of force applied on a body and displacement done by the body. So, in the vectors, it can be said that work done is the dot product of force applied on a body and displacement done by the body.
Mathematically, the above statement can be represented as –
$\vec W = \vec F.\vec s$
Putting the values of work done, force and displacement in the above equation –
$
\implies 16 = \left( {3\hat i - 2\hat j + \hat k} \right).\left( {2\hat i - 4\hat j + c\hat k} \right) \\
\implies 16 = 6 + 8 + c \\
$
Now, doing transposition method, we get –
$
\implies c = 16 - 14 \\
\implies c = 2 \\
$
Hence, the value of $c$ is $2$.
Therefore, the vector of displacement can be expressed as –
$\vec s = \left( {2\hat i - 4\hat j + 2\hat k} \right)m$
Hence, the correct option for this question is (D).
Note: The dot product of any orthogonal vector with itself is always one and the dot product of any orthogonal vector with any other orthogonal vector is always zero.
$
\hat i.\hat i = 1 \\
\hat j.\hat j = 1 \\
\hat k.\hat k = 1 \\
\hat i.\hat j = 0 \\
\hat i.\hat k = 0 \\
\hat j.\hat k = 0 \\
$
Let there be any two vectors such as –
$
\vec A = a\hat i + b\hat j + c\hat k \\
\vec B = x\hat i + y\hat j + z\hat k \\
$
Then, $\vec A.\vec B = ax + by + cz$
Complete step by step answer:
Let the force applied on a body be $\vec F$, displacement produced by the body be $\vec s$ and the work done by the body be $\vec W$.
According to the question, it is given that –
$\implies \vec F = \left( {3\hat i - 2\hat j + \hat k} \right)N$
$\implies \vec s = \left( {2\hat i - 4\hat j + c\hat k} \right)m$ and
$\implies \vec W = 16J$
Work done is the transfer of energy for the displacement of an object using the application of force. Work has only magnitude and no direction therefore, it is the scalar quantity. The S.I unit of work is Joule. It transfers energy from one place to another.
Now, we know that the work done can be defined as the product of force applied on a body and displacement done by the body. So, in the vectors, it can be said that work done is the dot product of force applied on a body and displacement done by the body.
Mathematically, the above statement can be represented as –
$\vec W = \vec F.\vec s$
Putting the values of work done, force and displacement in the above equation –
$
\implies 16 = \left( {3\hat i - 2\hat j + \hat k} \right).\left( {2\hat i - 4\hat j + c\hat k} \right) \\
\implies 16 = 6 + 8 + c \\
$
Now, doing transposition method, we get –
$
\implies c = 16 - 14 \\
\implies c = 2 \\
$
Hence, the value of $c$ is $2$.
Therefore, the vector of displacement can be expressed as –
$\vec s = \left( {2\hat i - 4\hat j + 2\hat k} \right)m$
Hence, the correct option for this question is (D).
Note: The dot product of any orthogonal vector with itself is always one and the dot product of any orthogonal vector with any other orthogonal vector is always zero.
$
\hat i.\hat i = 1 \\
\hat j.\hat j = 1 \\
\hat k.\hat k = 1 \\
\hat i.\hat j = 0 \\
\hat i.\hat k = 0 \\
\hat j.\hat k = 0 \\
$
Let there be any two vectors such as –
$
\vec A = a\hat i + b\hat j + c\hat k \\
\vec B = x\hat i + y\hat j + z\hat k \\
$
Then, $\vec A.\vec B = ax + by + cz$
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
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics 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)