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

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

Given:

• $\mathbf{u} = 8 \langle \cos 105^\circ, \sin 105^\circ \rangle$
• $\mathbf{v} = 4 \langle \cos 205^\circ, \sin 205^\circ \rangle$

We can calculate the components:

$\mathbf{u}_x = 8 \cos 105^\circ, \quad \mathbf{u}_y = 8 \sin 105^\circ$

$\mathbf{v}_x = 4 \cos 205^\circ, \quad \mathbf{v}_y = 4 \sin 205^\circ$

Step 2: Calculate the components

Using a calculator for the trigonometric functions:

$\cos 105^\circ \approx -0.2588, \quad \sin 105^\circ \approx 0.9659$ $\cos 205^\circ \approx -0.9063, \quad \sin 205^\circ \approx -0.4226$

Now, compute the components:

$\mathbf{u}_x = 8 \times (-0.2588) \approx -2.0704, \quad \mathbf{u}_y = 8 \times 0.9659 \approx 7.7272$

$\mathbf{v}_x = 4 \times (-0.9063) \approx -3.6252, \quad \mathbf{v}_y = 4 \times (-0.4226) \approx -1.6904$

Step 3: Find the sum of the vectors

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

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

$\mathbf{u} + \mathbf{v} = \langle -2.0704 - 3.6252, 7.7272 - 1.6904 \rangle$

$\mathbf{u} + \mathbf{v} \approx \langle -5.6956, 6.0368 \rangle$

Step 4: Calculate the magnitude of the resultant vector

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

$\|\mathbf{u} + \mathbf{v}\| = \sqrt{(-5.6956)^2 + (6.0368)^2}$

$\|\mathbf{u} + \mathbf{v}\| \approx \sqrt{32.4293 + 36.4430} \approx \sqrt{68.8723} \approx 8.3$

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

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

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

$\theta = \tan^{-1}\left(\frac{6.0368}{-5.6956}\right)$

$\theta \approx \tan^{-1}(-1.060) \approx -46.5^\circ$

Since the resultant vector is in the second quadrant (negative x-component and positive y-component), the direction angle should be adjusted:

$\theta \approx 180^\circ - 46.5^\circ \approx 134^\circ$

Final Answer:

• Magnitude: $8.3$ (rounded to the nearest tenth)
• Direction: $134^\circ$ (rounded to the nearest degree)

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

Related Questions:

1. How do you calculate the dot product of two vectors?
2. What is the significance of the angle between two vectors in vector operations?
3. Can you explain how to find the vector projection of one vector onto another?
4. How can vector cross products be applied in physics?
5. What are some practical applications of vector addition in engineering?

Tip:

When adding vectors graphically, always remember to consider the direction (angle) and magnitude to ensure accurate results.