To solve this, let's break down the problem step by step using vector components.

Step 1: Analyze the given vectors:

• Vector A: 50 km at 60° northeast
• Vector B: 20 km at 20° northeast

We need to calculate the components of these vectors in the x and y directions using trigonometry.

Step 2: Vector Components

1. Magnitude of the x component of vector A: The x-component of vector A is given by: $A_x = |A| \cdot \cos(\theta_A)$ where $|A| = 50 \, \text{km}$ and $\theta_A = 60^\circ$. $A_x = 50 \cdot \cos(60^\circ) = 50 \cdot 0.5 = 25.0 \, \text{km}$

2. Magnitude of the y component of vector A: The y-component of vector A is: $A_y = |A| \cdot \sin(\theta_A)$ $A_y = 50 \cdot \sin(60^\circ) = 50 \cdot 0.866 = 43.3 \, \text{km}$

3. Magnitude of the x component of vector B: The x-component of vector B is: $B_x = |B| \cdot \cos(\theta_B)$ where $|B| = 20 \, \text{km}$ and $\theta_B = 20^\circ$. $B_x = 20 \cdot \cos(20^\circ) = 20 \cdot 0.9397 = 18.8 \, \text{km}$

4. Magnitude of the y component of vector B: The y-component of vector B is: $B_y = |B| \cdot \sin(\theta_B)$ $B_y = 20 \cdot \sin(20^\circ) = 20 \cdot 0.3420 = 6.8 \, \text{km}$

Step 3: Resultant Vector C

Now, to find the overall magnitude and direction of vector C:

• The x-component of vector C is: $C_x = A_x + B_x = 25.0 + 18.8 = 43.8 \, \text{km}$

• The y-component of vector C is: $C_y = A_y + B_y = 43.3 + 6.8 = 50.1 \, \text{km}$

5. Magnitude of vector C: The magnitude of the resultant vector C is: $|C| = \sqrt{C_x^2 + C_y^2}$ $|C| = \sqrt{43.8^2 + 50.1^2} = \sqrt{1916.64 + 2510.01} = \sqrt{4426.65} \approx 66.5 \, \text{km}$

6. Direction of vector C: The angle above the horizontal (relative to the x-axis) is: $\theta_C = \tan^{-1}\left(\frac{C_y}{C_x}\right)$ $\theta_C = \tan^{-1}\left(\frac{50.1}{43.8}\right) \approx \tan^{-1}(1.14) = 48.6^\circ$

Final Answers:

1. x-component of vector A: $25.0 \, \text{km}$
2. y-component of vector A: $43.3 \, \text{km}$
3. x-component of vector B: $18.8 \, \text{km}$
4. y-component of vector B: $6.8 \, \text{km}$
5. Overall magnitude of vector C: $66.5 \, \text{km}$
6. Direction of vector C: $48.6^\circ$ above the horizontal

Let me know if you'd like further clarification, details, or if you have any questions!

Follow-up Questions:

1. How would the resultant change if vector B had a larger magnitude?
2. What if vector A had an angle of 45° instead of 60°?
3. How do you compute the components for a vector in 3D space?
4. Can you explain the significance of vector addition in physics?
5. How would you subtract two vectors geometrically?

Tip: Breaking vectors into components is one of the most useful tools for solving problems in physics and engineering, especially when dealing with forces or velocities!