A particles is acted upon by constant forces 4i + j - 3k and 3i + j - k which displace it from a poi

Q: Wait, why do we add the forces first? Can't we just calculate work for each force and add the results?

You absolutely can! The work done by the net force is equal to the sum of the work done by individual forces. However, in JEE Advanced, time is your currency. Adding the vectors first reduces the number of dot product operations, minimizing the chance of arithmetic errors. It is a more elegant and efficient path to the same destination.

Q: I always get confused with displacement. Is it $\vec{r}_{initial} - \vec{r}_{final}$ or the other way around?

Always remember: displacement is the change in position. It is final minus initial. Think of it as a vector pointing from where you started to where you ended. If you reverse it, you get a negative sign, which would flip the sign of your work done—a classic trap! Always visualize the arrow pointing from the start to the finish.

Q: Why does the dot product make the $\hat{i}, \hat{j}, \hat{k}$ components disappear?

Because the dot product is a projection. When you take the dot product of unit vectors like $\hat{i} \cdot \hat{i}$, you get 1, but $\hat{i} \cdot \hat{j}$ is 0. The cross-terms vanish, leaving you with a scalar value. Work is energy, and energy is a scalar quantity—it has magnitude but no direction. The dot product is the mathematical mechanism that strips away the directional information to leave us with the magnitude of energy transfer.

Question

A particles is acted upon by constant forces $4\hat{i} + \hat{j} - 3\hat{k}$ and $3\hat{i} + \hat{j} - \hat{k}$ which displace it from a point $\hat{i} + 2\hat{j} + 3\hat{k}$ to the point $5\hat{i} + 4\hat{j} + \hat{k}$. The work done in standard units by the forces is given by

Options:

(A) 15

(B) 30

(C) 25

(D) 40

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer. Today, we are not just solving a problem; we are observing the elegant choreography of forces in three-dimensional space. We have a particle moving from point $A$ to point $B$ under the influence of two distinct forces, $\vec{F}_1$ and $\vec{F}_2$.

Our mission is to calculate the total work done. This problem serves as a gateway to understanding how we translate physical intuition into the rigorous language of vector algebra.

### The Principle of Superposition

Imagine a particle floating in a void, pushed by two invisible hands. One hand pushes with force $\vec{F}_1 = 4\hat{i} + \hat{j} - 3\hat{k}$, and the other with $\vec{F}_2 = 3\hat{i} + \hat{j} - \hat{k}$.

Rather than calculating the work for each force individually, we utilize the **Principle of Superposition**. We replace these two forces with a single, equivalent net force, $\vec{F}_{net} = \vec{F}_1 + \vec{F}_2$.

Summing the components:
$$\vec{F}_{net} = (4+3)\hat{i} + (1+1)\hat{j} + (-3-1)\hat{k} = 7\hat{i} + 2\hat{j} - 4\hat{k}$$

### Defining the Journey

Next, we define the path of the particle. It starts at position $\vec{r}_A = \hat{i} + 2\hat{j} + 3\hat{k}$ and ends at $\vec{r}_B = 5\hat{i} + 4\hat{j} + \hat{k}$.

The displacement vector, $\vec{d}$, represents the change in position and is defined as $\vec{d} = \vec{r}_B - \vec{r}_A$. Precision is vital here to avoid errors with signs or subtraction order.

Calculating the displacement:
$$\vec{d} = (5-1)\hat{i} + (4-2)\hat{j} + (1-3)\hat{k} = 4\hat{i} + 2\hat{j} - 2\hat{k}$$

### The Bridge of the Dot Product

We now connect the net force and the displacement to find the work done. Work is defined as the scalar product of force and displacement: $W = \vec{F}_{net} \cdot \vec{d}$.

The dot product is the ideal tool because work is a scalar quantity. It projects the force onto the direction of displacement, effectively measuring how much of the force contributes to the movement.

### The Final Calculation

We perform the arithmetic by multiplying the corresponding components:
$$W = (7)(4) + (2)(2) + (-4)(-2)$$

Expanding this, we get:
$$W = 28 + 4 + 8$$

Summing these values, we arrive at the final result:
**$W = 40$ units of work.**

Notice how the unit vectors $\hat{i}, \hat{j}, \hat{k}$ vanished; they fulfilled their purpose by ensuring we only multiplied like-components. You have successfully navigated the vector space to reduce a complex, multi-dimensional problem to a single, beautiful scalar value.

Select Answer:

Visualized Solution

Visualizing the Physical Scenario

Forces Acting on the Particle

Resultant Force $F_{n e t}$

Substituting Force Vectors

Calculating $F_{n e t}$ Components

Displacement Vector $d$

Substituting Position Vectors

Calculating $d$ Components

Applying the Work Done Formula

Substituting Vectors for Dot Product

Calculating the Scalar Product

Final Numerical Calculation

Final Answer

The Sigma Insight: Scalar (Dot) Product

Analyzing the Setup

The Principle of Superposition

Defining the Journey

The Bridge of the Dot Product

The Final Calculation

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