Chapter 1, Problem 52P

(a)

To determine

The components of $\overrightarrow{E}$ in unit vector notation directed along positive x axis.

Answer to Problem 52P

The components of $\overrightarrow{E}$ in unit vector notation directed along positive x axis is $\overrightarrow{E}=\left(15.1\widehat{i}+7.72\widehat{j}\right)\text{cm}$.

Explanation of Solution

The figure 1 shows the vector notation of the following magnitudes.

Write the expression for the vectors $\overrightarrow{E}$ along x axis.

  $\overrightarrow{E}=E_1\cos \theta \widehat{i}+E_1\sin \theta \widehat{j}$        (I)

Here, $\overrightarrow{E}$ is the vector along x axis, $\theta$ is the direction of the vector, and $E_1$ is the vector component along x axis and y axis.

Conclusion:

Substitute $17.0\text{cm}$ for $E_1$ and $27.0°$ for $\theta$ in the equation (I) to find $\overrightarrow{E}$.

  $\begin{array}{c}\overrightarrow{E}=\left(17.0\text{cm}\right)\cos \left(27.0°\right)\widehat{i}+\left(17.0\text{cm}\right)\sin \left(27.0°\right)\widehat{j}\\ =\left(15.1\widehat{i}+7.72\widehat{j}\right)\text{cm}\end{array}$

Therefore, the components of $\overrightarrow{E}$ in unit vector notation directed along positive x axis is $\overrightarrow{E}=\left(15.1\widehat{i}+7.72\widehat{j}\right)\text{cm}$.

(b)

To determine

The components of $\overrightarrow{F}$ in unit vector notation directed along positive y axis.

Answer to Problem 52P

The components of $\overrightarrow{F}$ in unit vector notation directed along positive y axis is $\overrightarrow{F}=\left(-7.72\widehat{i}+15.1\widehat{j}\right)\text{cm}$.

Explanation of Solution

Write the expression for the vectors $\overrightarrow{F}$ along positive y axis.

  $\overrightarrow{F}=F_1\cos \left(90+\theta \right)\widehat{i}+F_1\sin \left(90+\theta \right)\widehat{j}$        (II)

Here, $\overrightarrow{F}$ is the vector along y axis, $\theta$ is the direction of the vector along positive x axis, and $F_1$ is the vector component along x axis and y axis.

Conclusion:

Substitute $17.0\text{cm}$ for $F_1$ and $27.0°$ for $\theta$ in the equation (II) to find $\overrightarrow{F}$.

  $\begin{array}{c}\overrightarrow{F}=\left(17.0\text{cm}\right)\cos \left(90°+27.0°\right)\widehat{i}+\left(17.0\text{cm}\right)\sin \left(90°+27.0°\right)\widehat{j}\\ =\left(17.0\text{cm}\right)\cos \left(117.0°\right)\widehat{i}+\left(17.0\text{cm}\right)\sin \left(117.0°\right)\widehat{j}\\ =\left(-7.72\widehat{i}+15.1\widehat{j}\right)\text{cm}\end{array}$

Therefore, the components of $\overrightarrow{F}$ in unit vector notation directed along positive y axis is $\overrightarrow{F}=\left(-7.72\widehat{i}+15.1\widehat{j}\right)\text{cm}$.

(c)

To determine

The components of $\overrightarrow{G}$ in unit vector notation directed along negative y axis.

Answer to Problem 52P

The components of $\overrightarrow{G}$ in unit vector notation directed along negative y axis is $\overrightarrow{G}=\left(-7.72\widehat{i}-15.1\widehat{j}\right)\text{cm}$.

Explanation of Solution

Write the expression for the vectors $\overrightarrow{G}$ along negative y axis.

  $\overrightarrow{G}=G_1\cos \left(\theta -270\right)\widehat{i}+G_1\sin \left(\theta -270\right)\widehat{j}$        (III)

Here, $\overrightarrow{G}$ is the vector along negative y axis, $\theta$ is the direction of the vector along negative y axis, and $G_1$ is the vector component along x axis and y axis.

Conclusion:

Substitute $17.0\text{cm}$ for $G_1$ and $27.0°$ for $\theta$ in the equation (III) to find $\overrightarrow{G}$.

  $\begin{array}{c}\overrightarrow{G}=\left(17.0\text{cm}\right)\cos \left(27.0°-270°+\right)\widehat{i}+\left(17.0\text{cm}\right)\sin \left(27.0°+270°\right)\widehat{j}\\ =\left(17.0\text{cm}\right)\cos \left(-243°\right)\widehat{i}+\left(17.0\text{cm}\right)\sin \left(243°\right)\widehat{j}\\ =\left(-7.72\widehat{i}-15.1\widehat{j}\right)\text{cm}\end{array}$

Therefore, the components of $\overrightarrow{G}$ in unit vector notation directed along negative y axis is $\overrightarrow{G}=\left(-7.72\widehat{i}-15.1\widehat{j}\right)\text{cm}$.