Question

A uniform force of $\left( {3\hat i + \hat j} \right)$ newton acts on a particle of mass $2kg$. Hence the particle is displaced from position $\left( {2\hat i + \hat k} \right)$ meter to position $\left( {4\hat i + 3\hat j - \hat k} \right)$meter. The work done by the force on the particle is
A. $15J$
B. $9J$
C. $6J$
D. $13J$

Answer 1

Hint: The dot product is used to measure how much the two vectors point in the same direction
A dot product gives a non-vector product for the multiplication of two vectors.
It is used to determine the vector magnitude in the direction of the vector which is projected and can determine when the two vectors are orthogonal or collinear.

Complete step by step answer:
The dot product helps to describe the lengths which is the length of a vector and square root of the dot product of the vector by itself
The dot product gives a non-vector product for the multiplication of the two vectors .It is like a scalar product of force and distance.
The given vectors are,
$\hat F = 3\hat i + \hat j$
We know that,
$\hat S = {\hat r_2} - {\hat r_1}$
Now after sustaining the r2 and r1 the S will be,
$\hat S = 2\hat i + 3\hat j - 2\hat k$
We know that work is,
 $W = \hat F.\hat S$
Now substitute the values in the above equation then the work will be,
$W = 6 + 3 + 0$
The work is,
$W = 9J$

So, the correct answer is “Option C”.

Note:
The inner product of a Euclidean space is called a dot product.
The sum of the products of the two sequences of numbers is known as the dot product.
Geometrically, it is defined as the product of the two vectors and the cosine angle between the two vectors of Euclidean magnitudes.