A force `vec(F) = hat(i) + 3hat(j) + 2hat(k)` acts on a particle to displace it from the point `A (hat(i) + 2hat(j) - 3hat(k))` to the point B `(3hat(i) - hat(j) + 5hat(k))`. The work done by the force will be

Options

A. 9 units
B. 7 units
C. 9 units
D. 10 units

Answer

To find the work done by the force \(\vec{F} = \hat{i} + 3\hat{j} + 2\hat{k}\) when displacing a particle from point \(A(\hat{i} + 2\hat{j} - 3\hat{k})\) to point \(B(3\hat{i} - \hat{j} + 5\hat{k})\), we can follow these steps:

### Step 1: Calculate the Displacement Vector \(\vec{r}\)
The displacement vector \(\vec{r}\) is given by the difference between the final position vector \(\vec{B}\) and the initial position vector \(\vec{A}\).

\[ \vec{r} = \vec{B} - \vec{A} \]

Substituting the position vectors:

\[ \vec{B} = 3\hat{i} - \hat{j} + 5\hat{k} \]
\[ \vec{A} = \hat{i} + 2\hat{j} - 3\hat{k} \]

Now, perform the subtraction:

\[ \vec{r} = (3\hat{i} - \hat{j} + 5\hat{k}) - (\hat{i} + 2\hat{j} - 3\hat{k}) \]
\[ = (3 - 1)\hat{i} + (-1 - 2)\hat{j} + (5 + 3)\hat{k} \]
\[ = 2\hat{i} - 3\hat{j} + 8\hat{k} \]

### Step 2: Calculate the Work Done \(W\)
The work done \(W\) by the force \(\vec{F}\) during the displacement \(\vec{r}\) is given by the dot product of the force vector and the displacement vector:

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

Substituting the vectors:

\[ \vec{F} = \hat{i} + 3\hat{j} + 2\hat{k} \]
\[ \vec{r} = 2\hat{i} - 3\hat{j} + 8\hat{k} \]

Now, calculate the dot product:

\[ W = (1)(2) + (3)(-3) + (2)(8) \]
\[ = 2 - 9 + 16 \]
\[ = 2 - 9 + 16 = 9 \]

### Final Answer
The work done by the force is \(W = 9\) units.

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