Given the vector u**u** equal to 4, open angle bracket, cosine, 295, degrees, comma, sine, 295, degrees, close angle bracket4⟨cos295∘,sin295∘⟩ and vector v**v** equal to 7, open angle bracket, cosine, 50, degrees, comma, sine, 50, degrees, close angle bracket, comma7⟨cos50∘,sin50∘⟩, find the sum u, plus, v**u**+**v** and write your answer in magnitude and direction form with the magnitude rounded to the nearest tenth and the direction rounded to the nearest degree

To find the sum of the vectors **u** and **v** and express it in magnitude and direction form, follow these steps:

### Step 1: Determine the components of the vectors **u** and **v**

Given:
- \(\mathbf{u} = 4 \langle \cos 295^\circ, \sin 295^\circ \rangle\)
- \(\mathbf{v} = 7 \langle \cos 50^\circ, \sin 50^\circ \rangle\)

We can calculate the components:

\[
\mathbf{u}_x = 4 \cos 295^\circ, \quad \mathbf{u}_y = 4 \sin 295^\circ
\]

\[
\mathbf{v}_x = 7 \cos 50^\circ, \quad \mathbf{v}_y = 7 \sin 50^\circ
\]

### Step 2: Calculate the components

Using a calculator for the trigonometric functions:

\[
\cos 295^\circ \approx 0.5736, \quad \sin 295^\circ \approx -0.8192
\]
\[
\cos 50^\circ \approx 0.6428, \quad \sin 50^\circ \approx 0.7660
\]

Now, compute the components:

\[
\mathbf{u}_x = 4 \times 0.5736 \approx 2.2944, \quad \mathbf{u}_y = 4 \times -0.8192 \approx -3.2768
\]

\[
\mathbf{v}_x = 7 \times 0.6428 \approx 4.4996, \quad \mathbf{v}_y = 7 \times 0.7660 \approx 5.3620
\]

### Step 3: Find the sum of the vectors

Add the corresponding components of \(\mathbf{u}\) and \(\mathbf{v}\):

\[
\mathbf{u} + \mathbf{v} = \langle \mathbf{u}_x + \mathbf{v}_x, \mathbf{u}_y + \mathbf{v}_y \rangle
\]

\[
\mathbf{u} + \mathbf{v} = \langle 2.2944 + 4.4996, -3.2768 + 5.3620 \rangle
\]

\[
\mathbf{u} + \mathbf{v} \approx \langle 6.7940, 2.0852 \rangle
\]

### Step 4: Calculate the magnitude of the resultant vector

The magnitude \( \|\mathbf{u} + \mathbf{v}\| \) is given by:

\[
\|\mathbf{u} + \mathbf{v}\| = \sqrt{(6.7940)^2 + (2.0852)^2}
\]

\[
\|\mathbf{u} + \mathbf{v}\| \approx \sqrt{46.1406 + 4.3480} \approx \sqrt{50.4886} \approx 7.1
\]

### Step 5: Determine the direction (angle) of the resultant vector

The direction \( \theta \) (relative to the positive x-axis) is given by:

\[
\theta = \tan^{-1}\left(\frac{\mathbf{u}_y + \mathbf{v}_y}{\mathbf{u}_x + \mathbf{v}_x}\right)
\]

\[
\theta = \tan^{-1}\left(\frac{2.0852}{6.7940}\right)
\]

\[
\theta \approx \tan^{-1}(0.3069) \approx 17^\circ
\]

### Final Answer:

- Magnitude: \( 7.1 \) (rounded to the nearest tenth)
- Direction: \( 17^\circ \) (rounded to the nearest degree)

Would you like to explore further details or have any questions?

### Related Questions:
1. How do you calculate the unit vector of a given vector?
2. What is the significance of vector magnitude in physics?
3. How can vectors be used to model real-world situations?
4. What is the process to resolve a vector into its components?
5. How do vector operations relate to complex numbers?

### Tip:
Always double-check the quadrant when calculating the direction of a vector sum to ensure the angle is correctly placed relative to the positive x-axis.

Explore how to find the sum of vectors u and v given in magnitude and direction form, rounded to the nearest tenth for magnitude and degree for direction. Follow step-by-step calculations and formulas for vector addition, magnitude calculation, and direction determination.

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