Chapter 3, Problem 2P

(a)

To determine

The magnitude and direction of $\overrightarrow{A}+\overrightarrow{B}$.

Answer to Problem 2P

The vector $\overrightarrow{A}+\overrightarrow{B}$ has magnitude $\underset{\_}{0.73\text{units}}$ and it is directed along the $\underset{\_}{+x\text{-direction}}$.

Explanation of Solution

Given that the vector $\overrightarrow{A}$ is directed along the positive $x$-axis and has magnitude $1.73\text{units}$, and the vector $\overrightarrow{B}$ is directed along the negative $x$-axis and has magnitude $1.00\text{unit}$.

Let us consider the $+x$-direction be to right.

The vector diagram of the vectors $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{A}+\overrightarrow{B}$ is shown in Figure 1.

Conclusion:

From the vector diagram in Figure 1 the magnitude of the resultant vector $\overrightarrow{A}+\overrightarrow{B}$ is obtained as,

$\begin{array}{c}\overrightarrow{A}+\overrightarrow{B}=1.73\text{units}-1.00\text{unit}\\ =0.73\text{units}\end{array}$

As shown in the vector diagram, the direction of $\overrightarrow{A}+\overrightarrow{B}$ is in the $+x$-direction.

Therefore, the vector $\overrightarrow{A}+\overrightarrow{B}$ has magnitude $\underset{\_}{0.73\text{units}}$ and it is directed along the $\underset{\_}{+x\text{-direction}}$.

(b)

To determine

The magnitude and direction of $\overrightarrow{A}-\overrightarrow{B}$.

Answer to Problem 2P

The vector $\overrightarrow{A}-\overrightarrow{B}$ has magnitude $\underset{\_}{2.73\text{units}}$ and it is directed along the $\underset{\_}{+x\text{-direction}}$.

Explanation of Solution

Given that the vector $\overrightarrow{A}$ is directed along the positive $x$-axis and has magnitude $1.73\text{units}$, and the vector $\overrightarrow{B}$ is directed along the negative $x$-axis and has magnitude $1.00\text{unit}$.

Let us consider the $+x$-direction be to right.

The vector diagram of the vectors $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{A}-\overrightarrow{B}$ is shown in Figure 2.

Conclusion:

From the vector diagram in Figure 2 the magnitude of the resultant vector $\overrightarrow{A}-\overrightarrow{B}$ is obtained as,

$\begin{array}{c}\overrightarrow{A}-\overrightarrow{B}=1.73\text{units}+1.00\text{unit}\\ =2.73\text{units}\end{array}$

As shown in the vector diagram, the direction of $\overrightarrow{A}-\overrightarrow{B}$ is in the $+x$-direction.

Therefore, the vector $\overrightarrow{A}-\overrightarrow{B}$ has magnitude $\underset{\_}{2.73\text{units}}$ and it is directed along the $\underset{\_}{+x\text{-direction}}$.

(c)

To determine

The magnitude and direction of $\overrightarrow{B}-\overrightarrow{A}$.

Answer to Problem 2P

The vector $\overrightarrow{B}-\overrightarrow{A}$ has magnitude $\underset{\_}{2.73\text{units}}$ and it is directed along the $\underset{\_}{-x\text{-direction}}$.

Explanation of Solution

Given that the vector $\overrightarrow{A}$ is directed along the positive $x$-axis and has magnitude $1.73\text{units}$, and the vector $\overrightarrow{B}$ is directed along the negative $x$-axis and has magnitude $1.00\text{unit}$.

Let us consider the $+x$-direction be to right.

The vector diagram of the vectors $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{B}-\overrightarrow{A}$ is shown in Figure 3.

Conclusion:

From the vector diagram in Figure 3 the magnitude of the resultant vector $\overrightarrow{B}-\overrightarrow{A}$ is obtained as,

$\begin{array}{c}\overrightarrow{B}-\overrightarrow{A}=1.00\text{unit}+1.73\text{units}\\ =2.73\text{units}\end{array}$

As shown in the vector diagram, the direction of $\overrightarrow{B}-\overrightarrow{A}$ is in the $-x$-direction.

Therefore, the vector $\overrightarrow{B}-\overrightarrow{A}$ has magnitude $\underset{\_}{2.73\text{units}}$ and it is directed along the $\underset{\_}{-x\text{-direction}}$.