Chapter 3, Problem 14P

Solution

To Determine:The xand y components of the vector $\overrightarrow{V}$ .

Solution: The x -component $V_x$ must be written as: $V_x=\left|\overrightarrow{V}\right|\sin \varphi$ and the y -component $V_y$ must be written as: $V_y=\left|\overrightarrow{V}\right|\cos \varphi$

Explanation:

Given:

The vector $\overrightarrow{V}$ makes an angle $\varphi$ with respect to the y-axis.

Formula Used:

The x -component $V_x$ of vector $\overrightarrow{V}$ is defined as:

  $V_x=\left|\overrightarrow{V}\right|\cos \theta$

The y-component of $V_y$ vector $\overrightarrow{V}$ is defined as:

  $V_y=\left|\overrightarrow{V}\right|\sin \theta$

Where:

• $\left|\overrightarrow{V}\right|$ is the magnitude of the vector $\overrightarrow{V}$
• $\theta$ is the direction of the vector  $\overrightarrow{V}$ measured respect to the positive x -axis counter-clockwise.

Calculations:

  $\theta =90-\varphi$

The x -component $V_x$ of vector $\overrightarrow{V}$ is defined as:

  $\begin{array}{l}{V}_x=\left|\overrightarrow{V}\right|\cos \theta \\ V_x=\left|\overrightarrow{V}\right|\cos \left(90-\varphi \right)\\ V_x=\left|\overrightarrow{V}\right|\sin \varphi \end{array}$

The y-component of $V_y$ vector $\overrightarrow{V}$ is defined as:

  $\begin{array}{l}{V}_y=\left|\overrightarrow{V}\right|\sin \theta \\ V_y=\left|\overrightarrow{V}\right|\sin \left(90-\varphi \right)\\ V_y=\left|\overrightarrow{V}\right|\cos \varphi \end{array}$

Conclusion:

Thex -component $V_x$ must be written as: $V_x=\left|\overrightarrow{V}\right|\sin \varphi$ and the y -component $V_y$ must be written as: $V_y=\left|\overrightarrow{V}\right|\cos \varphi$ .