Chapter 1, Problem 74P

(a)

To determine

The maximum and minimum of $\overrightarrow{A}+\overrightarrow{B}$.

Answer to Problem 74P

The maximum and minimum of $\overrightarrow{A}+\overrightarrow{B}$ are $29\text{m}$ and $5\text{m}$ respectively.

Explanation of Solution

When the two vectors are in the same direction, $\overrightarrow{A}+\overrightarrow{B}$ will be maximum. The maximum of $\overrightarrow{A}+\overrightarrow{B}$ is given by,

    ${\left(\overrightarrow{A}+\overrightarrow{B}\right)}_{\max }=\left|\overrightarrow{A}\right|+\left|\overrightarrow{B}\right|$

When the two vectors are in the opposite direction, $\overrightarrow{A}+\overrightarrow{B}$ will be minimum. The minimum of $\overrightarrow{A}+\overrightarrow{B}$ is given by,

    ${\left(\overrightarrow{A}+\overrightarrow{B}\right)}_{\min }=\left|\overrightarrow{B}\right|-\left|\overrightarrow{A}\right|$

Conclusion:

Substitute 12 m for $\left|\overrightarrow{A}\right|$ and 17 m for $\left|\overrightarrow{B}\right|$ to get ${\left(\overrightarrow{A}+\overrightarrow{B}\right)}_{\max }$.

    $\begin{array}{c}{\left(\overrightarrow{A}+\overrightarrow{B}\right)}_{\max }=\left(12\text{m}\right)+\left(17\text{m}\right)\\ =29\text{m}\end{array}$

Substitute 12 m for $\left|\overrightarrow{A}\right|$ and 17 m for $\left|\overrightarrow{B}\right|$ to get ${\left(\overrightarrow{A}+\overrightarrow{B}\right)}_{\min }$.

    $\begin{array}{c}{\left(\overrightarrow{A}+\overrightarrow{B}\right)}_{\min }=\left(17\text{m}\right)-\left(12\text{m}\right)\\ =5\text{m}\end{array}$

Therefore, the maximum and minimum of $\overrightarrow{A}+\overrightarrow{B}$ are $29\text{m}$ and $5\text{m}$ respectively.

(b)

To determine

The maximum and minimum of $\overrightarrow{A}-\overrightarrow{B}$.

Answer to Problem 74P

The maximum and minimum of $\overrightarrow{A}-\overrightarrow{B}$ are $29\text{m}$ and $5\text{m}$ respectively.

Explanation of Solution

When the two vectors are in the opposite direction, $\overrightarrow{A}-\overrightarrow{B}$ will be maximum. The maximum of $\overrightarrow{A}-\overrightarrow{B}$ is given by,

    ${\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\max }=\left|\overrightarrow{A}\right|-\left(-\left|\overrightarrow{B}\right|\right)=\left|\overrightarrow{A}\right|+\left|\overrightarrow{B}\right|$

When the two vectors are in the same direction, $\overrightarrow{A}-\overrightarrow{B}$ will be minimum. The minimum of $\overrightarrow{A}-\overrightarrow{B}$ is given by,

    ${\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\min }=\left|\overrightarrow{B}\right|-\left|\overrightarrow{A}\right|$

Conclusion:

Substitute 12 m for $\left|\overrightarrow{A}\right|$ and 17 m for $\left|\overrightarrow{B}\right|$ to get ${\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\max }$.

    $\begin{array}{c}{\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\max }=\left(12\text{m}\right)-\left(-17\text{m}\right)\\ =29\text{m}\end{array}$

Substitute 12 m for $\left|\overrightarrow{A}\right|$ and 17 m for $\left|\overrightarrow{B}\right|$ to get ${\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\min }$.

    $\begin{array}{c}{\left(\overrightarrow{A}-\overrightarrow{B}\right)}_{\min }=\left(17\text{m}\right)-\left(12\text{m}\right)\\ =5\text{m}\end{array}$

Therefore, the maximum and minimum of $\overrightarrow{A}-\overrightarrow{B}$ are $29\text{m}$ and $5\text{m}$ respectively.