To solve for the components of a vector [tex]\(P\)[/tex] given its magnitude and direction, let's start by summarizing the given information:
1. The magnitude of the vector [tex]\(P\)[/tex] is [tex]\(P = 4.5 \, \text{m}\)[/tex].
2. The direction of the vector [tex]\(P\)[/tex] makes an angle [tex]\(\theta = 36^\circ\)[/tex] with the positive [tex]\(x\)[/tex]-axis.
### Step-by-Step Solution:
To find the components [tex]\(P_x\)[/tex] and [tex]\(P_y\)[/tex] along the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-axes respectively, we can use trigonometric functions:
1. Convert the angle from degrees to radians:
Angles in trigonometric functions are typically measured in radians. To convert degrees to radians, we use the formula:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
Plugging in our angle [tex]\(\theta = 36^\circ\)[/tex]:
[tex]\[
\theta_\text{rad} = 36 \times \frac{\pi}{180} \approx 0.6283 \, \text{radians}
\][/tex]
2. Calculate the [tex]\(x\)[/tex]-component [tex]\(P_x\)[/tex]:
The [tex]\(x\)[/tex]-component of the vector is found using the cosine function:
[tex]\[
P_x = P \cos(\theta)
\][/tex]
Plugging in the values:
[tex]\[
P_x = 4.5 \cos(0.6283) \approx 3.6406 \, \text{m}
\][/tex]
3. Calculate the [tex]\(y\)[/tex]-component [tex]\(P_y\)[/tex]:
The [tex]\(y\)[/tex]-component of the vector is found using the sine function:
[tex]\[
P_y = P \sin(\theta)
\][/tex]
Plugging in the values:
[tex]\[
P_y = 4.5 \sin(0.6283) \approx 2.6450 \, \text{m}
\][/tex]
### Summary of the Components:
- The x-component [tex]\(P_x\)[/tex] of the vector [tex]\(P = 4.5 \, \text{m}\)[/tex] at an angle [tex]\(\theta = 36^\circ\)[/tex] is approximately [tex]\(3.6406 \, \text{m}\)[/tex].
- The y-component [tex]\(P_y\)[/tex] of the vector [tex]\(P = 4.5 \, \text{m}\)[/tex] at an angle [tex]\(\theta = 36^\circ\)[/tex] is approximately [tex]\(2.6450 \, \text{m}\)[/tex].
So, the components of the vector [tex]\(P\)[/tex] are:
[tex]\[
P_x \approx 3.6406 \, \text{m}
\][/tex]
[tex]\[
P_y \approx 2.6450 \, \text{m}
\][/tex]
