To determine Pac-Man's initial position vector, we'll start with the relation between the initial position vector [tex]\(\vec{r}_i\)[/tex], the displacement vector [tex]\(\overrightarrow{\Delta r}\)[/tex], and the final position vector [tex]\(\vec{r}_f\)[/tex]:
[tex]\[
\vec{r}_i + \overrightarrow{\Delta r} = \vec{r}_f
\][/tex]
We can rearrange this equation to solve for the initial position vector, [tex]\(\vec{r}_i\)[/tex]:
[tex]\[
\vec{r}_i = \vec{r}_f - \overrightarrow{\Delta r}
\][/tex]
Given:
- Displacement vector: [tex]\(\overrightarrow{\Delta r} = 2.0 \hat{i} - 3.0 \hat{j} + 6.0 \hat{k}\)[/tex]
- Final position vector: [tex]\(\vec{r}_f = 3.0 \hat{j} - 4.0 \hat{k}\)[/tex]
(Noting that the final position in the [tex]\(x\)[/tex]-direction is 0, we can write [tex]\(\vec{r}_f\)[/tex] as [tex]\(0 \hat{i} + 3.0 \hat{j} - 4.0 \hat{k}\)[/tex]).
We subtract the displacement vector from the final position vector component-wise:
[tex]\[
\vec{r}_i = \left(0 \hat{i} + 3.0 \hat{j} - 4.0 \hat{k}\right) - \left(2.0 \hat{i} - 3.0 \hat{j} + 6.0 \hat{k}\right)
\][/tex]
This gives us individual components for [tex]\(\vec{r}_i\)[/tex]:
[tex]\[
r_x = 0 - 2.0 = -2.0
\][/tex]
[tex]\[
r_y = 3.0 - (-3.0) = 3.0 + 3.0 = 6.0
\][/tex]
[tex]\[
r_z = -4.0 - 6.0 = -10.0
\][/tex]
Therefore, Pac-Man's initial position vector is:
[tex]\[
\vec{r}_i = -2.0 \hat{i} + 6.0 \hat{j} - 10.0 \hat{k}
\][/tex]
So, the initial position vector components are:
[tex]\[
r_x = -2.0, \quad r_y = 6.0, \quad r_z = -10.0
\][/tex]
