Answer

Verified

350.5k+ views

**Hint:**Try to recall how you would express the work done as a dot product of the above two quantities. Also remember that the only contributing vector components are the non-zero components, both in magnitude and direction. In other words, focus only on the component of the force acting along the z-direction while computing your dot product.

**Formula used:**

Work done $W = \vec{F}\; .\vec{S}$ where F is the force vector and S is the displacement vector.

**Complete answer:**

Let us first establish that a vector in 3-dimensions can be broken into 3 components:

The x-axis component $\hat{i}$,

The y-axis component $\hat{j}$, and

The z-axis component $\hat{k}$.

Each component of a vector depicts the magnitude of influence of that vector in a given direction. The $\hat{i}$, $\hat{j}$ and $\hat{k}$ represent unit vectors in the x-, y- and z-direction respectively, and the number that precedes them represent the magnitude of the vector in that direction.

Now, we have a body that can move only along the direction of the z-axis. This means that any distance that we take that this body covers will be in the z($\vec{k}$)-direction. Therefore, the distance that the body travels under the influence of the force can be represented by the displacement vector $\vec{S} = 0\hat{i}+0\hat{j}+4\hat{k}$.

The work done by the force $\vec{F} = -1\hat{i}+2\hat{j}+3\hat{k}$ to move the body by a distance $\vec{S} =4\hat{k}$ is given as the scalar product of the two, i.e.,:

$W = \vec{F}.\vec{S} = \left(-1\hat{i}+2\hat{j}+3\hat{k}\right). \left(4\hat{k}\right)$

$\Rightarrow W = \left(3\hat{k}\right). \left(4\hat{k}\right) = 12\;J$

Therefore, only the z-component of the force contributes to moving the body in the z-direction. Thus, the work done by the force in moving the body through a distance of $4\;m$ is $12\;J$

**Note:**

Remember that the dot product of two vectors results in a scalar quantity and hence is it not directional. Another form of expressing the dot product when instead of the individual components the angle $\theta$ between the two vectors is given is:

$W = \vec{F}.\vec{S} = |F||S|cos\theta$

In the above problem, we consider only $ W = \left(3\hat{k}\right). \left(4\hat{k}\right) $, which means $ W = 4 \times 3 \cos 0^{\circ} = 12\;J$ since $cos0^{\circ} =1$ . This is the same reason why we do not consider $\hat{i}.\hat{j}$ or $\hat{j}.\hat{k}$ or $\hat{i}.\hat{k}$ since for them, $\theta =90^{\circ} \Rightarrow cos 90^{\circ} = 0, \Rightarrow W=0$.

Therefore, the work done is numerically quantified only when the vectors are not perpendicular to each other and the vectors have non-zero components.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

The provincial president of the constituent assembly class 11 social science CBSE

Gersoppa waterfall is located in AGuyana BUganda C class 9 social science CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The hundru falls is in A Chota Nagpur Plateau B Calcutta class 8 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE