Question 1191444: B. Magnitude of Vectors (Trigonometry)
3. Find the magnitude and direction of the sum w of the following vectors. Express the direction as a measure of the angle between v and w.
- If ||u|| = 6.5, ||v|| = 7.2 and the angle between them measures 150 degrees.
Show full solution, thank you!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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°
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