Let's break down the problem and solve it step by step.

1. Understanding the Problem:

• Emman is trying to find a key using vectors on a coordinate plane, where the center of the basketball court is the origin (0, 0).
• He has three vector clues:
  • A = 20 meters at 30° North of East.
  • B = (-10 meters, 20 meters).
  • C = (-20 meters, -40 meters).

2. Vector A:

Vector A is given in polar coordinates. We need to convert this to Cartesian coordinates.

• Magnitude of A = 20 meters.
• Direction of A = 30° North of East.

The Cartesian coordinates for a vector are calculated as:

• $A_x = A \cdot \cos(\theta)$
• $A_y = A \cdot \sin(\theta)$

So,

• $A_x = 20 \times \cos(30°) = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3}$ meters ≈ 17.32 meters.
• $A_y = 20 \times \sin(30°) = 20 \times \frac{1}{2} = 10$ meters.

Thus, A in Cartesian coordinates is approximately $(17.32, 10)$.

3. Vector B:

Vector B is already given in Cartesian form:

• B = (-10, 20) meters.

4. Vector C:

Vector C is also given in Cartesian form:

• C = (-20, -40) meters.

5. Summing the Vectors:

To find the final position where the key is located, we sum the vectors A, B, and C:

$\text{Total } x = A_x + B_x + C_x$ $\text{Total } y = A_y + B_y + C_y$

Substituting the values:

• $\text{Total } x = 17.32 + (-10) + (-20)$
• $\text{Total } y = 10 + 20 + (-40)$

Calculating:

• $\text{Total } x = 17.32 - 10 - 20 = -12.68$ meters.
• $\text{Total } y = 10 + 20 - 40 = -10$ meters.

6. Final Position:

The key is located at approximately (-12.68 meters, -10 meters) relative to the origin (the center of the basketball court).

Would you like more details or have any questions?

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Follow-up Questions:

1. What is the significance of converting polar coordinates to Cartesian coordinates?
2. How do you calculate the direction of a vector given its components?
3. What is the importance of vector addition in physics and real-world applications?
4. How can you determine the magnitude and direction of the resulting vector?
5. What is the role of trigonometric functions in vector calculations?
6. How do different coordinate systems affect vector calculations?
7. Can you represent the final position graphically to visualize the key's location?
8. How would the problem change if the direction in vector A was different?

Tip:

When dealing with vectors, always make sure to carefully analyze the direction and magnitude before proceeding with calculations to ensure accuracy.