A body moves from a position `vec(r_(1))=(2hati-3hatj-4hatk)` m to a position `vec(r_(2))=(3hati-4hatj+5hatk)m` under the influence of a constant force `vecF=(4hati+hatj+6hatk)N`. The work done by the force is :

Options

A. 60 J
B. 59 J
C. 58 J
D. 57 J

Answer

To find the work done by the force on the body moving from position \(\vec{r_1} = (2\hat{i} - 3\hat{j} - 4\hat{k})\) m to position \(\vec{r_2} = (3\hat{i} - 4\hat{j} + 5\hat{k})\) m under the influence of a constant force \(\vec{F} = (4\hat{i} + \hat{j} + 6\hat{k})\) N, we can follow these steps:

### Step 1: Calculate the Displacement Vector
The displacement vector \(\vec{d}\) can be calculated by subtracting the initial position vector \(\vec{r_1}\) from the final position vector \(\vec{r_2}\):

\[ \vec{d} = \vec{r_2} - \vec{r_1} \]

Substituting the values:

\[ \vec{d} = (3\hat{i} - 4\hat{j} + 5\hat{k}) - (2\hat{i} - 3\hat{j} - 4\hat{k}) \]

Calculating the components:

\[ \vec{d} = (3 - 2)\hat{i} + (-4 + 3)\hat{j} + (5 + 4)\hat{k} \]
\[ \vec{d} = 1\hat{i} - 1\hat{j} + 9\hat{k} \]

### Step 2: Calculate the Work Done
The work done \(W\) by the force \(\vec{F}\) on the displacement \(\vec{d}\) is given by the dot product:

\[ W = \vec{F} \cdot \vec{d} \]

Substituting the values of \(\vec{F}\) and \(\vec{d}\):

\[ W = (4\hat{i} + \hat{j} + 6\hat{k}) \cdot (1\hat{i} - 1\hat{j} + 9\hat{k}) \]

Calculating the dot product:

\[ W = 4 \cdot 1 + 1 \cdot (-1) + 6 \cdot 9 \]
\[ W = 4 - 1 + 54 \]
\[ W = 57 \text{ Joules} \]

### Final Answer
The work done by the force is \(57\) Joules.

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