To determine the magnitude and direction of the vector sum [tex]\( \mathbf{u} + \mathbf{v} + \mathbf{w} \)[/tex], let's go through the solution step-by-step:
1. Understand the Given Vectors:
- Vector [tex]\( \mathbf{u} \)[/tex]
- Magnitude: 57.817
- Direction: [tex]\(19^\circ\)[/tex]
- Vector [tex]\( \mathbf{v} \)[/tex]
- Magnitude: 57.817
- Direction: [tex]\(161^\circ\)[/tex]
- Vector [tex]\( \mathbf{w} \)[/tex]
- Magnitude: 92.326
- Direction: [tex]\(11^\circ\)[/tex]
2. Convert each vector to its rectangular (Cartesian) coordinates:
To do this, we use the formulas for converting from polar to rectangular coordinates:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
where [tex]\( r \)[/tex] is the magnitude and [tex]\( \theta \)[/tex] is the direction.
For vector [tex]\( \mathbf{u} \)[/tex]:
[tex]\[
u_x = 57.817 \cos(19^\circ)
\][/tex]
[tex]\[
u_y = 57.817 \sin(19^\circ)
\][/tex]
For vector [tex]\( \mathbf{v} \)[/tex]:
[tex]\[
v_x = 57.817 \cos(161^\circ)
\][/tex]
[tex]\[
v_y = 57.817 \sin(161^\circ)
\][/tex]
For vector [tex]\( \mathbf{w} \)[/tex]:
[tex]\[
w_x = 92.326 \cos(11^\circ)
\][/tex]
[tex]\[
w_y = 92.326 \sin(11^\circ)
\][/tex]
3. Add the rectangular coordinates to get the resultant vector's coordinates:
[tex]\[
\text{Resultant}_x = u_x + v_x + w_x
\][/tex]
[tex]\[
\text{Resultant}_y = u_y + v_y + w_y
\][/tex]
4. Convert the resultant back to polar coordinates:
Use the formulas for converting from rectangular to polar coordinates:
[tex]\[
\text{Magnitude} = \sqrt{x^2 + y^2}
\][/tex]
[tex]\[
\text{Direction} = \tan^{-1}\left(\frac{y}{x}\right)
\][/tex]
Therefore:
[tex]\[
\text{Resultant Magnitude} = \sqrt{(\text{Resultant}_x)^2 + (\text{Resultant}_y)^2}
\][/tex]
[tex]\[
\text{Resultant Direction} = \tan^{-1}\left(\frac{\text{Resultant}_y}{\text{Resultant}_x}\right)
\][/tex]
5. Round the answers:
- Round the magnitude to the thousandths place.
- Round the direction to the nearest degree.
By following these steps, the magnitude and direction of [tex]\( \mathbf{u} + \mathbf{v} + \mathbf{w} \)[/tex] are found to be:
- Magnitude: 106.150
- Direction: 31 degrees
So, the final resultant vector [tex]\( \mathbf{u} + \mathbf{v} + \mathbf{w} \)[/tex] has a magnitude of 106.150 (rounded to the thousandths) and a direction of 31 degrees (rounded to the nearest degree).
