https://file.aiquickdraw.com/mathbot/file/172709491531943mlaxdg.jpg

Let's analyze and break down the problems from the image.

### Problem 1:
For each vector polygon in the images (a), (b), (c), and (d), we are asked to write the vector equation that gives the resultant vector \( R \).

- **Figure (a)**: The vectors form a closed polygon, so the vector equation is:
  \[
  \vec{O} + \vec{A} + \vec{B} + \vec{C} + \vec{D} = \vec{R}
  \]
  (We may need to adjust signs depending on directions in further detail.)

- **Figure (b)**: Similar approach, the vectors should sum up to give \( R \):
  \[
  \vec{O} + \vec{A} + \vec{B} + \vec{C} + \vec{D} = \vec{R}
  \]

- **Figure (c)**: Here we have three vectors summing up to give the resultant:
  \[
  \vec{O} + \vec{M} + \vec{N} + \vec{P} = \vec{R}
  \]

- **Figure (d)**: The vectors combine similarly:
  \[
  \vec{O} + \vec{M} + \vec{N} + \vec{P} = \vec{R}
  \]

---

### Problem 2:
You're given a vector \( A \) with a magnitude of 20 units and an angle of \( 130^\circ \) with two other vectors, \( B \) and \( C \). 

- \( B \) has an angle of \( 80^\circ \)
- \( C \) has an angle of \( 210^\circ \)

Using vector decomposition, you can break down vector \( A \) using trigonometric relations (sine and cosine rules) and solve for the magnitudes of \( B \) and \( C \).

---

### Problem 3:
You're given vector \( T \) with a magnitude of 50 units and an angle of \( 120^\circ \), along with vectors \( R \) and \( S \), where:
- \( R \) has a magnitude of 30 units,
- \( S \) has a magnitude of 66 units.

You would use the law of cosines and sines to resolve the angles and magnitudes of \( R \) and \( S \).

---

### Problem 4:
Vector \( A \) has a magnitude of 50 units and an angle of \( 210^\circ \), decomposing into vectors \( B \) and \( C \), where:
- \( B \)'s magnitude is unknown and angle is to be calculated,
- \( C \) has a magnitude of 37.5 units and an angle of \( 75^\circ \).

We can apply vector decomposition to find the unknown magnitude and angle of \( B \) using trigonometric rules.

---

Would you like detailed solutions for any of these problems, including step-by-step vector decomposition and solving the system of equations?

### Here are some related questions:
1. What is the general formula for adding vectors in a polygon?
2. How do you resolve a vector into its horizontal and vertical components?
3. What is the Law of Cosines, and how does it apply to vector addition?
4. How can we calculate the magnitude of the resultant vector when multiple vectors are given?
5. How do we interpret angles in vector problems when they're measured from different reference points?

### Tip:
Always sketch the vector diagram clearly and label angles properly before applying vector addition rules. It helps to avoid confusion, especially when dealing with different reference directions.

For each vector polygon in the diagram, write the vector equation that gives the resultant vector R.

This problem involves writing vector equations for polygons to determine the resultant vector R. It covers vector addition concepts, including trigonometric vector decomposition. Suitable for students in Grades 11-12.

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