To find the sum of vectors [tex]\(u\)[/tex] and [tex]\(v\)[/tex] and express it in terms of magnitude and direction, we will follow these steps:
1. Decompose Each Vector into Components:
- Vector [tex]\(u\)[/tex] has a magnitude of 2 and an angle of 95 degrees.
- [tex]\(u_x = 2 \cos 95^\circ \)[/tex]
- [tex]\(u_y = 2 \sin 95^\circ \)[/tex]
- Vector [tex]\(v\)[/tex] has a magnitude of 4 and an angle of 165 degrees.
- [tex]\(v_x = 4 \cos 165^\circ \)[/tex]
- [tex]\(v_y = 4 \sin 165^\circ \)[/tex]
2. Compute the Components of Each Vector:
The components (x and y) for these vectors are:
[tex]\[
u_x = 2 \cos 95^\circ \\
u_y = 2 \sin 95^\circ \\
v_x = 4 \cos 165^\circ \\
v_y = 4 \sin 165^\circ
\][/tex]
3. Sum the Corresponding Components:
Add the x-components and y-components separately:
[tex]\[
\text{resultant}_x = u_x + v_x \\
\text{resultant}_y = u_y + v_y \\
\][/tex]
4. Calculate the Magnitude of the Resultant Vector:
Use the Pythagorean theorem to find the magnitude [tex]\( R \)[/tex]:
[tex]\[
R = \sqrt{\text{resultant}_x^2 + \text{resultant}_y^2}
\][/tex]
5. Find the Angle of the Resultant Vector:
Use the arctangent function to find the angle [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \arctan\left(\frac{\text{resultant}_y}{\text{resultant}_x}\right)
\][/tex]
Make sure the angle is in the correct quadrant based on the signs of [tex]\(\text{resultant}_x\)[/tex] and [tex]\(\text{resultant}_y\)[/tex], and convert from radians to degrees if necessary.
6. Normalize the Angle:
Ensure that [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex].
After performing these calculations, we find:
- The magnitude of the resultant vector [tex]\( R \)[/tex] is approximately [tex]\(5.0\)[/tex], rounded to the nearest tenth.
- The direction of the resultant vector [tex]\( \theta \)[/tex] is approximately [tex]\(143^\circ\)[/tex], rounded to the nearest degree.
Therefore, the sum [tex]\( u + v \)[/tex] expressed in magnitude and direction form is:
[tex]\[
\boxed{5.0 \text{ units, direction } 143^\circ}
\][/tex]
