To analyze the vector \( \mathbf{a} \) with a magnitude of 7 m/s and an angle of \( 0^\circ \) using the analytical method, we'll break down the vector into its components. Here's how:

### 1. Understanding the Vector
- **Magnitude**: \( |\mathbf{a}| = 7 \, \text{m/s} \)
- **Angle**: \( \theta = 0^\circ \)

### 2. Decompose the Vector
The vector can be decomposed into its horizontal (x) and vertical (y) components using trigonometric functions.

- **Horizontal Component (\( \mathbf{a}_x \))**:
  \[
  \mathbf{a}_x = |\mathbf{a}| \cdot \cos(\theta)
  \]
  Since \( \theta = 0^\circ \), \( \cos(0^\circ) = 1 \):
  \[
  \mathbf{a}_x = 7 \, \text{m/s} \cdot 1 = 7 \, \text{m/s}
  \]

- **Vertical Component (\( \mathbf{a}_y \))**:
  \[
  \mathbf{a}_y = |\mathbf{a}| \cdot \sin(\theta)
  \]
  Since \( \theta = 0^\circ \), \( \sin(0^\circ) = 0 \):
  \[
  \mathbf{a}_y = 7 \, \text{m/s} \cdot 0 = 0 \, \text{m/s}
  \]

### 3. Final Vector Components
- \( \mathbf{a}_x = 7 \, \text{m/s} \)
- \( \mathbf{a}_y = 0 \, \text{m/s} \)

### 4. Vector Representation
The vector \( \mathbf{a} \) can now be represented as:
\[
\mathbf{a} = (7 \, \text{m/s}, 0 \, \text{m/s})
\]

This indicates that the vector \( \mathbf{a} \) is entirely along the x-axis, with no component along the y-axis.

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Would you like more details or have any questions?

Here are some related questions you might consider:

1. What happens to the vector components if the angle changes to \( 90^\circ \)?
2. How do you calculate the magnitude of a vector given its components?
3. What is the significance of the angle in vector decomposition?
4. How do you find the direction of a vector from its components?
5. Can you explain the relationship between vector components and trigonometric functions?
6. How do you represent a vector graphically on a coordinate plane?
7. What are unit vectors, and how are they related to vector decomposition?
8. How do vectors add together analytically?

**Tip:** When working with vectors, always pay attention to the direction and magnitude, as they determine the vector's overall effect.

Understand how to analyze vector a with a magnitude of 7 m/s and an angle of 0 degrees using the analytical method. Explore the decomposition of the vector into horizontal and vertical components through trigonometric functions.

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