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

where [tex]\(\vec{F}\)[/tex] is applied force, and [tex]\(\vec{d}\)[/tex] is displacement which is given by

[tex]\[
\begin{aligned}
\vec{d} & = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} + (z_2 - z_1) \hat{k} \\
& = 2 \hat{i} + (-1) \hat{j} + 4 \hat{k} \\
& = 2 \hat{i} - \hat{j} + 4 \hat{k}
\end{aligned}
\][/tex]

[tex]\[
\vec{F} = 4 \hat{i} - 3 \hat{j} + 2 \hat{k}
\][/tex]

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

[tex]\[
\begin{aligned}
W & = (4 \hat{i} - 3 \hat{j} + 2 \hat{k}) \cdot (2 \hat{i} - \hat{j} + 4 \hat{k}) \\
& = (4 \cdot 2) + (-3 \cdot -1) + (2 \cdot 4) \\
& = 8 + 3 + 8 \\
& = 19
\end{aligned}
\][/tex]

So,

[tex]\[
W = 19
\][/tex]



Answer :

To solve for the work [tex]\(W\)[/tex] done by the force [tex]\(\vec{F}\)[/tex] on the displacement [tex]\(\vec{d}\)[/tex], follow these steps:

1. Identify the given vectors:

The force vector [tex]\(\vec{F}\)[/tex] is given by:
[tex]\[ \vec{F} = 4\vec{i} - 3\vec{j} + 2\vec{k} \][/tex]

The displacement vector [tex]\(\vec{d}\)[/tex] is given by:
[tex]\[ \vec{d} = -\vec{i} - 3\vec{j} + 5\vec{k} \][/tex]

2. Recall the dot product formula:

The work [tex]\(W\)[/tex] done by the force over the displacement is given by the dot product of the force and displacement vectors:
[tex]\[ W = \vec{F} \cdot \vec{d} \][/tex]

3. Calculate the dot product:

The dot product of two vectors [tex]\(\vec{A} = a_1\vec{i} + a_2\vec{j} + a_3\vec{k}\)[/tex] and [tex]\(\vec{B} = b_1\vec{i} + b_2\vec{j} + b_3\vec{k}\)[/tex] is given by:
[tex]\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \][/tex]

For [tex]\(\vec{F} = 4\vec{i} - 3\vec{j} + 2\vec{k}\)[/tex] and [tex]\(\vec{d} = -\vec{i} - 3\vec{j} + 5\vec{k}\)[/tex], compute the dot product:
[tex]\[ W = (4)(-1) + (-3)(-3) + (2)(5) \][/tex]

4. Perform the arithmetic:

- The product of the [tex]\(\vec{i}\)[/tex] components: [tex]\(4 \times -1 = -4\)[/tex]
- The product of the [tex]\(\vec{j}\)[/tex] components: [tex]\(-3 \times -3 = 9\)[/tex]
- The product of the [tex]\(\vec{k}\)[/tex] components: [tex]\(2 \times 5 = 10\)[/tex]

Adding these results together:
[tex]\[ W = -4 + 9 + 10 \][/tex]

5. Sum the results:

[tex]\[ W = 15 \][/tex]

Thus, the work done by the force [tex]\(\vec{F}\)[/tex] over the displacement [tex]\(\vec{d}\)[/tex] is:
[tex]\[ W = 15 \][/tex]