Question

Figure 4-22 shows the initial position i and the final position f of a particle. What are the (a) initial position vector$\overrightarrow{{\mathbf{r}}_{\mathbf{1}}}$and (b) final position vector$\overrightarrow{{\mathbf{r}}_{\mathbf{f}}}$, both in unit-vector notation? (c) What is the x component of displacement$\mathbf{∆}\overrightarrow{\mathbf{r}}$?

Short answer

a) Initial position vector$\overrightarrow{r_i}=7\stackrel{⏜}{\mathrm{i}}+\stackrel{⏜}{\mathrm{j}}-2\stackrel{⏜}{\mathrm{k}}$

b) Final position vector $\overrightarrow{r_f}=5\stackrel{⏜}{\mathrm{i}}-3\stackrel{⏜}{\mathrm{j}}+\stackrel{⏜}{\mathrm{k}}$

c) The x component of displacement$∆\overrightarrow{r}=-2\stackrel{⏜}{i}$

Step-by-step solution

Given information

$$\begin{gathered} r_{ix}=7\hspace{0.33em}m \\ {r}_{iy}=1\hspace{0.33em}m \\ {r}_{iz}=-2\hspace{0.33em}m \\ {r}_{fx}=5\hspace{0.33em}m \\ {r}_{fy}=-3\hspace{0.33em}m \\ {r}_{fz}=1\hspace{0.33em}m \end{gathered}$$

To understand the concept

Vector can be written by measuring the distances along each direction. Displacement can be calculated by change in position between two vectors.

Formulae:

Unit vector notation

$\overrightarrow{r}=r_x\stackrel{⏜}{i}+r_y\stackrel{⏜}{j}+r_z\stackrel{⏜}{k}$

(a) To find initial position vector r1→

Initial position vector $\left(\overrightarrow{r_1}\right)$

$$\begin{gathered} \overrightarrow{r_1}=r_{ix}\stackrel{⏜}{i}+r_{iy}\stackrel{⏜}{j}+r_{iz}\stackrel{⏜}{k} \\ \overrightarrow{r_1}=7\stackrel{⏜}{i}+\stackrel{⏜}{j}-2\stackrel{⏜}{k} \end{gathered}$$

(b) To find initial position vector rf→

$$\begin{gathered} \overrightarrow{r_f}=r_{fx}\stackrel{⏜}{i}+r_{fy}\stackrel{⏜}{j}+r_{fz}\stackrel{⏜}{k} \\ \overrightarrow{r_f}=5\stackrel{⏜}{i}-3\stackrel{⏜}{j}+\stackrel{⏜}{k} \end{gathered}$$

(c) To find x component of displacement ∆r→

We have, displacement vector is given by:

$$\begin{gathered} ∆\overrightarrow{r}=\overrightarrow{r_f}-\overrightarrow{r_i} \\ =\left(5-7\right)\stackrel{⏜}{i}+\left(-3-1\right)\stackrel{⏜}{j}+\left(1-\left(-2\right)\right)\stackrel{⏜}{k} \\ =-2\stackrel{⏜}{i}-4\stackrel{⏜}{j}+3\stackrel{⏜}{k} \end{gathered}$$

Hence,

The x component of the displacement is $∆\overrightarrow{r}=-2\stackrel{⏜}{i}$.