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The problem shows a sailboat race course with four displacement vectors \( \vec{A} \), \( \vec{B} \), \( \vec{C} \), and \( \vec{D} \), with their magnitudes given as:

- \( A = 3.30 \, \text{km} \),
- \( B = 5.30 \, \text{km} \),
- \( C = 5.20 \, \text{km} \).

The course is a closed polygon, so the final displacement vector \( \vec{D} \) brings the boat back to the starting point. The angles between the vectors are also given: \( 40^\circ \) between \( \vec{B} \) and \( \vec{C} \), and \( 23.0^\circ \) between \( \vec{C} \) and \( \vec{D} \).

The goal is to find:

1. The magnitude of the final leg \( \vec{D} \),
2. The angle \( \theta \) between \( \vec{A} \) and \( \vec{D} \).

For part (a), the magnitude of \( \vec{D} \), you have already calculated it to be \( 7.2 \, \text{km} \), which appears correct.

Now, let's move on to part (b): solving for the angle \( \theta \). I will use the vector geometry and apply trigonometry to calculate this angle.

Let me do the calculations for you.The angle \( \theta \) between vectors \( \vec{A} \) and \( \vec{D} \) is approximately \( 43.2^\circ \).

Let me know if you want further clarifications or have any other questions.

Here are five related questions to extend your understanding:

1. How does the law of cosines help in solving problems involving non-right triangles?
2. What is the significance of using radians in trigonometric calculations?
3. How can vector components be used to solve this problem instead of the law of cosines?
4. What other methods can be applied to find the final displacement in closed-loop problems?
5. How does the relationship between vectors change when the course shape is not a polygon?

**Tip:** When working with angles in trigonometric functions, always ensure the correct unit (degrees vs radians) for accurate results.

A sailboat race course consists of four legs, defined by the displacement vectors A, B, C, and D as the drawing indicates. The magnitudes of the first three vectors are A = 3.30 km, B = 5.30 km, and C = 5.20 km. Using the data in the drawing, find (a) the distance of the fourth leg, and (b) the angle θ.

Solve a sailboat race course problem involving four displacement vectors. Use trigonometric principles like the Law of Cosines and Law of Sines to calculate the distance of the fourth leg and the angle θ. Detailed solution provided for students in Grades 10-12.

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