using that image solve for each compenet form of each vector

To find the component form of each vector in the image, we'll break down each vector into its horizontal (x) and vertical (y) components.

### Analyzing the Vectors:

1. **Blue Vector**:
   - The vector appears to start from the origin (0, 0) and extends to a point in the second quadrant.
   - From the graph, we can estimate its endpoint coordinates:
     - **x-component**: The vector moves leftward from the origin, which gives a negative x-component.
     - **y-component**: The vector moves upward from the origin, giving a positive y-component.
   - Endpoint of the blue vector seems to be at \((-3, 4)\).

2. **Red Vector**:
   - This vector also seems to start from the origin and extends to a point in the fourth quadrant.
   - We can estimate the endpoint coordinates:
     - **x-component**: The vector moves rightward from the origin, giving a positive x-component.
     - **y-component**: The vector moves downward from the origin, giving a negative y-component.
   - Endpoint of the red vector seems to be at \((4, -2)\).

### Component Form of Each Vector:

Given the vectors start at the origin (0, 0), the component form of a vector is simply the coordinates of its endpoint.

- **Blue Vector**: 
  - Endpoint: \((-3, 4)\)
  - Component form: \(\mathbf{v}_1 = \langle -3, 4 \rangle\)

- **Red Vector**:
  - Endpoint: \((4, -2)\)
  - Component form: \(\mathbf{v}_2 = \langle 4, -2 \rangle\)

### Summary:

- **Blue Vector**: \(\mathbf{v}_1 = \langle -3, 4 \rangle\)
- **Red Vector**: \(\mathbf{v}_2 = \langle 4, -2 \rangle\)

These components tell you how much each vector moves in the x-direction (horizontally) and the y-direction (vertically).

Would you like more details on these calculations, or do you have any further questions? Here are some related questions to deepen your understanding:

1. How do you calculate the magnitude of each vector from its components?
2. How do you determine the direction (angle) of each vector?
3. What would the resultant vector be if you added these two vectors?
4. How would you subtract these two vectors, and what would the result look like?
5. How do you calculate the dot product of these two vectors?

**Tip**: Always remember that the sign of each component (positive or negative) is determined by the direction of the vector along the x and y axes.

Explore how to find the component form of vectors using coordinate geometry. Learn about the x and y components of vectors and their endpoint coordinates. Suitable for high school students studying vectors and coordinate geometry.

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