Chapter 1, Problem 47P

(a)

To determine

The vector sum of $\overrightarrow{A}+\overrightarrow{B}$.

Answer to Problem 47P

The vector sum of $\overrightarrow{A}+\overrightarrow{B}$ is $2\widehat{i}-6\widehat{j}$.

Explanation of Solution

Write the expression for the given vector sum.

  $\overrightarrow{A}+\overrightarrow{B}$        (I)

Conclusion:

Substitute $\left(3\widehat{i}-2\widehat{j}\right)$ for $\overrightarrow{A}$ and $\left(-1\widehat{i}-4\widehat{j}\right)$ for $\overrightarrow{B}$ in the equation (I) to find $\overrightarrow{A}+\overrightarrow{B}$.

  $\begin{array}{c}\overrightarrow{A}+\overrightarrow{B}=\left(3\widehat{i}-2\widehat{j}\right)+\left(-1\widehat{i}-4\widehat{j}\right)\\ =2\widehat{i}-6\widehat{j}\end{array}$

Therefore, the vector sum of $\overrightarrow{A}+\overrightarrow{B}$ is $2\widehat{i}-6\widehat{j}$.

(b)

To determine

The vector difference of $\overrightarrow{A}-\overrightarrow{B}$.

Answer to Problem 47P

The vector difference of $\overrightarrow{A}-\overrightarrow{B}$ is $4\widehat{i}+2\widehat{j}$.

Explanation of Solution

Write the expression for the given vector difference.

  $\overrightarrow{A}-\overrightarrow{B}$        (II)

Conclusion:

Substitute $\left(3\widehat{i}-2\widehat{j}\right)$ for $\overrightarrow{A}$ and $\left(-1\widehat{i}-4\widehat{j}\right)$ for $\overrightarrow{B}$ in the equation (II) to find $\overrightarrow{A}-\overrightarrow{B}$.

  $\begin{array}{c}\overrightarrow{A}-\overrightarrow{B}=\left(3\widehat{i}-2\widehat{j}\right)-\left(-1\widehat{i}-4\widehat{j}\right)\\ =3\widehat{i}-2\widehat{j}+1\widehat{i}+4\widehat{j}\\ =4\widehat{i}+2\widehat{j}\end{array}$

Therefore, the vector difference of $\overrightarrow{A}-\overrightarrow{B}$ is $4\widehat{i}+2\widehat{j}$.

(c)

To determine

The magnitude of sum of the vectors $\left|\overrightarrow{A}+\overrightarrow{B}\right|$.

Answer to Problem 47P

The magnitude of sum of the vectors $\left|\overrightarrow{A}+\overrightarrow{B}\right|$ is $6.32$.

Explanation of Solution

Write the expression for the magnitude of the vector sum.

  $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{{\text{A}}_x^2+{\text{B}}_y^2}$        (III)

Here, $\left|\overrightarrow{A}+\overrightarrow{B}\right|$ is the magnitude of the vector, ${\text{A}}_x$ and ${\text{B}}_y$ are the x and y components of the vector.

Conclusion:

Substitute $2\widehat{i}$ for ${\text{A}}_x$ and $-6\widehat{j}$ for ${\text{B}}_y$ in equation (III) to find $\left|\overrightarrow{A}+\overrightarrow{B}\right|$.

  $\begin{array}{c}\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{{\left(2\right)}^2+{\left(-6\right)}^2}\\ =6.32\end{array}$

Therefore, the magnitude of sum of the vectors $\left|\overrightarrow{A}+\overrightarrow{B}\right|$ is $6.32$.

(d)

To determine

The magnitude of difference of the vectors $\left|\overrightarrow{A}-\overrightarrow{B}\right|$.

Answer to Problem 47P

The magnitude of difference of the vectors $\left|\overrightarrow{A}-\overrightarrow{B}\right|$ is $4.47$.

Explanation of Solution

Write the expression for the magnitude of the vector difference.

  $\left|\overrightarrow{A}-\overrightarrow{B}\right|=\sqrt{{\text{A}}_x^2-{\text{B}}_y^2}$        (IV)

Here, $\left|\overrightarrow{A}-\overrightarrow{B}\right|$ is the magnitude of the vector, ${\text{A}}_x$ and ${\text{B}}_y$ are the x and y components of the vector.

Conclusion:

Substitute $4\widehat{i}$ for ${\text{A}}_x$ and $2\widehat{j}$ for ${\text{B}}_y$ in equation (IV) to find $\left|\overrightarrow{A}-\overrightarrow{B}\right|$.

  $\begin{array}{c}\left|\overrightarrow{A}-\overrightarrow{B}\right|=\sqrt{{\left(4\right)}^2+{\left(2\right)}^2}\\ =4.47\end{array}$

Therefore, the magnitude of sum of the vectors $\left|\overrightarrow{A}-\overrightarrow{B}\right|$ is $4.47$.

(e)

To determine

The direction of the vectors $\left|\overrightarrow{A}+\overrightarrow{B}\right|$ and $\left|\overrightarrow{A}-\overrightarrow{B}\right|$.

Answer to Problem 47P

The direction of the vectors $\left|\overrightarrow{A}+\overrightarrow{B}\right|$ is $288°$ and the direction of vectors $\left|\overrightarrow{A}-\overrightarrow{B}\right|$ is $26.6°$.

Explanation of Solution

Write the expression for the direction of the vector $\left|\overrightarrow{A}+\overrightarrow{B}\right|$.

  ${\theta }_{\left|\overrightarrow{A}+\overrightarrow{B}\right|}={\tan }^{-1}\left(\frac{{\text{B}}_y}{{\text{A}}_x}\right)$        (V)

Write the expression for the direction of the vector $\left|\overrightarrow{A}-\overrightarrow{B}\right|$.

  ${\theta }_{\left|\overrightarrow{A}-\overrightarrow{B}\right|}={\tan }^{-1}\left(\frac{{\text{B}}_y}{{\text{A}}_x}\right)$        (VI)

Conclusion:

Substitute $2\widehat{i}$ for ${\text{A}}_x$ and $-6\widehat{j}$ for ${\text{B}}_y$ in equation (V) to find ${\theta }_{\left|\overrightarrow{A}+\overrightarrow{B}\right|}$.

  $\begin{array}{c}{\theta }_{\left|\overrightarrow{A}+\overrightarrow{B}\right|}={\tan }^{-1}\left(\frac{-6}{2}\right)\\ =-71.6°\\ {\theta }_{\left|\overrightarrow{A}+\overrightarrow{B}\right|}=360°-71.6°\\ =288°\end{array}$

Substitute $4\widehat{i}$ for ${\text{A}}_x$ and $2\widehat{j}$ for ${\text{B}}_y$ in equation (VI) to find ${\theta }_{\left|\overrightarrow{A}-\overrightarrow{B}\right|}$.

  $\begin{array}{c}{\theta }_{\left|\overrightarrow{A}-\overrightarrow{B}\right|}={\tan }^{-1}\left(\frac{2}{4}\right)\\ =26.6°\end{array}$

Therefore, the direction of the vectors $\left|\overrightarrow{A}+\overrightarrow{B}\right|$ is $288°$ and the direction of vectors $\left|\overrightarrow{A}-\overrightarrow{B}\right|$ is $26.6°$