Question

A force $\overrightarrow{F} = 5\widehat{i} + 6\widehat{j} + 4\widehat{k}$ acting on a body, produces displacement $\overrightarrow S = 6\widehat{i} - 5\widehat{k}$. Work done by the force is
A. $10$ units
B. $18$ units
C. $11$ units
D. $5$ units

Answer 1

Hint: In this question we have to find the work done by the force if the force of $\overrightarrow{F} = 5\widehat{i} + 6\widehat{j} + 4\widehat{k}$ is acting on a body producing a displacement of $\overrightarrow S = 6\widehat{i} - 5\widehat{k}$. We will be using the known formula for work done i.e., work done is the dot product of Force acting and displacement produced.

Formula Used:
Following formulae is useful for solving this question
$W = \overrightarrow{F} \cdot \vec S \\ \widehat{i} \cdot \widehat{j} = 0 \\ \widehat{i} \cdot \widehat{k} = 0 \\ \widehat{j} \cdot \widehat{k} = 0$
$\widehat{i} \cdot \widehat{i} = 1 \\ \widehat{j} \cdot \widehat{j} = 1 \\ \widehat{k} \cdot \widehat{k} = 1 $

Complete step by step solution:
We are given force applied and displacement produced as $\overrightarrow{F} = 5\widehat{i} + 6\widehat{j} + 4\widehat{k}$ and $\overrightarrow S = 6\widehat{i} - 5\widehat{k}$.
We know that work done is the dot product of force applied and displacement produced.
By using formula $W = \overrightarrow{F} \cdot \vec S$, we get
$W = \overrightarrow{F} \cdot \vec S \\ W = (5\widehat{i} + 6\widehat{j} + 4\widehat{k}) \cdot (6\widehat{i} - 5\widehat{k})$
On further solving, we get
$W = 5\widehat{i} \cdot 6\widehat{i} + 5\widehat{i} \cdot ( - 5\widehat{k}) + 6\widehat{j} \cdot 6\widehat{i} + 6\widehat{j} \cdot ( - 5\widehat{k}) + 4\widehat{k} \cdot 6\widehat{i} + 4\widehat{k} \cdot ( - 5\widehat{k}) \\ W = 30 \cdot \widehat{i} \cdot \widehat{i} - 25 \cdot \widehat{i} \cdot \widehat{k} + 36 \cdot \widehat{i} \cdot \widehat{j} - 30 \cdot \widehat{j} \cdot \widehat{k} + 24 \cdot \widehat{i} \cdot \widehat{k} - 20 \cdot \widehat{k} \cdot \widehat{k} = 30 - 20$
On further solving, we get
$W = 10$ units

Option ‘A’ is correct

Additional information : The "work done'' by the force is the scalar product of the force vector and the relegation vector of the object. We say that the force "does work'' if it's wielded while the object moves (has a relegation vector) and in such a way that the scalar product of the force and relegation vectors is non-zero. In drugs, a force is an influence that can change the stir of an object. A force can get an object with mass to change its haste, i.e., to accelerate. Force can also be described intimately as a drive or a pull. A force has both magnitude and direction, making it a vector volume.

Note: While performing the dot product of unit vectors $\widehat{i},\widehat{j},\widehat{k}$ two vectors must have the same length.
The dot product is calculated using the formula
$a \cdot b= \sum_{i=1}^n a_i b_i$