Solved Examples of Vector Addition Using Trigonometry

📚 Quick Study Guide

🔍 Vector Components: Resolve vectors into horizontal ($x$) and vertical ($y$) components using trigonometry: $A_x = A \cos(\theta)$, $A_y = A \sin(\theta)$. ➕ Adding Components: Add the $x$ components of all vectors to get the resultant $x$ component ($R_x$) and similarly for the $y$ components ($R_y$). 📐 Resultant Magnitude: Find the magnitude of the resultant vector ($R$) using the Pythagorean theorem: $R = \sqrt{R_x^2 + R_y^2}$. 🧭 Resultant Angle: Determine the direction (angle $\theta$) of the resultant vector using the inverse tangent function: $\theta = \tan^{-1}(\frac{R_y}{R_x})$. Pay attention to the quadrant! 💡 Important Tip: Always draw a diagram to visualize the vectors and their components. This helps in avoiding errors in sign conventions and angle calculations.

🧪 Practice Quiz

1. A force of 10 N acts at an angle of 30° to the horizontal. What is the horizontal component of the force?
   1. 5 N
   2. 8.66 N
   3. 10 N
   4. 11.55 N
2. Two forces act on an object: 5 N at 0° and 8 N at 90°. What is the magnitude of the resultant force?
   1. 3 N
   2. 13 N
   3. $\sqrt{39}$ N
   4. $\sqrt{89}$ N
3. A displacement vector has components $x = 3$ m and $y = 4$ m. What is the angle it makes with the x-axis?
   1. 36.87°
   2. 45°
   3. 53.13°
   4. 60°
4. Vector A has a magnitude of 6 and points in the +x direction. Vector B has a magnitude of 8 and points in the +y direction. What is the magnitude of A + B?
   1. 2
   2. 10
   3. 14
   4. 48
5. A plane flies with a velocity of 200 km/h due east. A wind blows with a velocity of 50 km/h due north. What is the magnitude of the plane's resultant velocity?
   1. 150 km/h
   2. 206.16 km/h
   3. 250 km/h
   4. 300 km/h
6. A vector has a magnitude of 10 units and makes an angle of 120° with the positive x-axis. What is its y-component?
   1. 5
   2. -5
   3. 8.66
   4. -8.66
7. Two forces, 3N and 4N, act on an object. The angle between the forces is 90 degrees. What is the direction of the resultant force with respect to the 3N force?
   1. 36.87°
   2. 45°
   3. 53.13°
   4. 60°