The work done by the force `vec( F ) = 6 hat(i) + 2 hat(j)` N in displacing an object from` vec( r_(1)) = 3 hat(i) + 8 hat( j)` to `vec( r_(2)) = 5 hat( i) - 4 hat(j)` m , is

Options

A. (a)12 J
B. (b)`-36 J`
C. (c)36J
D. (d)`-12J`

Answer

To find the work done by the force \(\vec{F} = 6 \hat{i} + 2 \hat{j}\) N in displacing an object from \(\vec{r_1} = 3 \hat{i} + 8 \hat{j}\) m to \(\vec{r_2} = 5 \hat{i} - 4 \hat{j}\) m, we can follow these steps:

### Step 1: Calculate the Displacement Vector
The displacement vector \(\vec{dr}\) can be calculated using the formula:
\[ \vec{dr} = \vec{r_2} - \vec{r_1} \]
Substituting the given vectors:
\[ \vec{dr} = (5 \hat{i} - 4 \hat{j}) - (3 \hat{i} + 8 \hat{j}) \]
Now, perform the subtraction:
\[ \vec{dr} = (5 - 3) \hat{i} + (-4 - 8) \hat{j} = 2 \hat{i} - 12 \hat{j} \]

### Step 2: Calculate the Work Done
The work done \(W\) by the force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{dr} \]
Substituting the values of \(\vec{F}\) and \(\vec{dr}\):
\[ W = (6 \hat{i} + 2 \hat{j}) \cdot (2 \hat{i} - 12 \hat{j}) \]
Calculating the dot product:
\[ W = (6 \cdot 2) + (2 \cdot -12) = 12 - 24 = -12 \text{ J} \]

### Conclusion
The work done by the force in displacing the object is \(-12\) Joules.

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