QUESTION

# The work done by the force $\overset{\to }{\mathop{F}}\,=\overset{\to }{\mathop{i}}\,+\overset{\to }{\mathop{j}}\,+\overset{\to }{\mathop{k}}\,$ acting on a particle is displaced from A (3, 3, 3) to the point B (4, 4, 4) is,(a) 2 units(b) 3 units(c) 4 units(d) 7 units

Hint: First calculate the distance by using the formula $\overset{\to }{\mathop{AB}}\,=\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{a}}\,$ and then use the formula $W=\overset{\to }{\mathop{F}}\,.\overset{\to }{\mathop{AB}}\,$ to get the value of work done. Use the formula $\overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{q}}\,=\left( \overset{\to }{\mathop{ai}}\,+\overset{\to }{\mathop{bj}}\,+\overset{\to }{\mathop{ck}}\, \right).\left( \overset{\to }{\mathop{di}}\,+\overset{\to }{\mathop{ej}}\,+\overset{\to }{\mathop{fk}}\, \right)=ad+be+cf$ to calculate the dot product.

To find out the work done by the given force we should write the given equation first, therefore,
$\overset{\to }{\mathop{F}}\,=\overset{\to }{\mathop{i}}\,+\overset{\to }{\mathop{j}}\,+\overset{\to }{\mathop{k}}\,$ ……………………………………….. (1)
Also, A (3, 3, 3) and B (4, 4, 4) ……………………………………… (2)
As we have given the two points A and B therefore it’s position vector will become,
$\overset{\to }{\mathop{a}}\,=3\overset{\to }{\mathop{i}}\,+3\overset{\to }{\mathop{j}}\,+3\overset{\to }{\mathop{k}}\,$ and $\overset{\to }{\mathop{b}}\,=4\overset{\to }{\mathop{i}}\,+4\overset{\to }{\mathop{j}}\,+4\overset{\to }{\mathop{k}}\,$ ……………………………………… (3)
The distance between two points is given by the formula, $\overset{\to }{\mathop{AB}}\,=\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{a}}\,$ therefore,
$\overset{\to }{\mathop{AB}}\,=\overset{\to }{\mathop{b}}\,-\overset{\to }{\mathop{a}}\,$
If we put the values of equation (3) in the above formula we will get,
$\therefore \overset{\to }{\mathop{AB}}\,=\left( 4\overset{\to }{\mathop{i}}\,+4\overset{\to }{\mathop{j}}\,+4\overset{\to }{\mathop{k}}\, \right)-\left( 3\overset{\to }{\mathop{i}}\,+3\overset{\to }{\mathop{j}}\,+3\overset{\to }{\mathop{k}}\, \right)$
Further simplification in the above equation will give,
$\therefore \overset{\to }{\mathop{AB}}\,=4\overset{\to }{\mathop{i}}\,+4\overset{\to }{\mathop{j}}\,+4\overset{\to }{\mathop{k}}\,-3\overset{\to }{\mathop{i}}\,-3\overset{\to }{\mathop{j}}\,-3\overset{\to }{\mathop{k}}\,$
By rearranging the above equation we will get,
$\therefore \overset{\to }{\mathop{AB}}\,=\left( 4\overset{\to }{\mathop{i}}\,-3\overset{\to }{\mathop{i}}\, \right)+\left( 4\overset{\to }{\mathop{j}}\,-3\overset{\to }{\mathop{j}}\, \right)+\left( 4\overset{\to }{\mathop{k}}\,-3\overset{\to }{\mathop{k}}\, \right)$
$\therefore \overset{\to }{\mathop{AB}}\,=\overset{\to }{\mathop{i}}\,+\overset{\to }{\mathop{j}}\,+\overset{\to }{\mathop{k}}\,$ ………………………………………………………. (4)
To proceed further in the solution we should know the formula of work done given below,
Formula:
$W=\overset{\to }{\mathop{F}}\,.\overset{\to }{\mathop{d}}\,$ Where $\overset{\to }{\mathop{d}}\,$ is the distance covered which is $\overset{\to }{\mathop{AB}}\,$ in this case,
Therefore the work by the given force is given by,
$W=\overset{\to }{\mathop{F}}\,.\overset{\to }{\mathop{AB}}\,$
If we put the values of equation (1) and equation (4) in the above equation we will get,
$W=\left( \overset{\to }{\mathop{i}}\,+\overset{\to }{\mathop{j}}\,+\overset{\to }{\mathop{k}}\, \right).\left( \overset{\to }{\mathop{i}}\,+\overset{\to }{\mathop{j}}\,+\overset{\to }{\mathop{k}}\, \right)$
Now to proceed further in the solution we should know the the formula given below,
Formula:
If $\overset{\to }{\mathop{p}}\,=\overset{\to }{\mathop{ai}}\,+\overset{\to }{\mathop{bj}}\,+\overset{\to }{\mathop{ck}}\,$ and $\overset{\to }{\mathop{q}}\,=\overset{\to }{\mathop{di}}\,+\overset{\to }{\mathop{ej}}\,+\overset{\to }{\mathop{fk}}\,$ then their dot product is given by, $\overset{\to }{\mathop{p}}\,.\overset{\to }{\mathop{q}}\,=\left( \overset{\to }{\mathop{ai}}\,+\overset{\to }{\mathop{bj}}\,+\overset{\to }{\mathop{ck}}\, \right).\left( \overset{\to }{\mathop{di}}\,+\overset{\to }{\mathop{ej}}\,+\overset{\to }{\mathop{fk}}\, \right)=ad+be+cf$.
By using the above formula in ‘W’ we will get,
$\therefore W=1\times 1+1\times 1+1\times 1$
$\therefore W=1+1+1$
Therefore, W = 3 units.
Therefore the work done by a given force from point A to point B is equal to 3 units.
Therefore the correct answer is option (b).

Note: You can also solve this problem by calculating $\left| \overset{\to }{\mathop{F}}\, \right|$ and distance AB by using distance formula and then using the formula $W=\left| \overset{\to }{\mathop{F}}\, \right|.AB$ to get the quick answer.