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]
