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.