Question 1191444
Here's how to find the magnitude and direction of the sum of the vectors u and v:

**1. Find the x and y components of each vector:**

* **Vector u:**
    * u_x = ||u|| * cos(θ_u) = 6.5 * cos(θ_u)
    * u_y = ||u|| * sin(θ_u) = 6.5 * sin(θ_u)

* **Vector v:**
    * v_x = ||v|| * cos(θ_v) = 7.2 * cos(θ_v)
    * v_y = ||v|| * sin(θ_v) = 7.2 * sin(θ_v)

We need to establish a reference angle. Let's assume vector *u*'s angle θ_u is 0 degrees. Then vector *v*'s angle θ_v is 150 degrees.

* **Vector u:**
    * u_x = 6.5 * cos(0°) = 6.5
    * u_y = 6.5 * sin(0°) = 0

* **Vector v:**
    * v_x = 7.2 * cos(150°) = 7.2 * (-√3/2) ≈ -6.24
    * v_y = 7.2 * sin(150°) = 7.2 * (1/2) = 3.6

**2. Find the x and y components of the resultant vector w:**

* w_x = u_x + v_x = 6.5 + (-6.24) ≈ 0.26
* w_y = u_y + v_y = 0 + 3.6 = 3.6

**3. Find the magnitude of w:**

* ||w|| = sqrt(w_x² + w_y²) = sqrt(0.26² + 3.6²) ≈ sqrt(0.0676 + 12.96) ≈ sqrt(13.0276) ≈ 3.61

**4. Find the direction of w:**

* θ_w = arctan(w_y / w_x) = arctan(3.6 / 0.26) ≈ arctan(13.85) ≈ 85.9°

Since w_x is positive and w_y is positive, the angle is in the first quadrant, so 85.9° is correct.

**5. Find the angle between u and w:**

The angle between u and w is simply θ_w - θ_u = 85.9° - 0° = 85.9°.

**Answers:**

* **Magnitude of w:** Approximately 3.61
* **Direction of w (relative to u):** Approximately 85.9°