Which vector equation below does NOT represent the line passing through the points  A(-1,-6), B(3, 3) ?

To determine which vector equation does not represent the line passing through the points \( A(-1, -6) \) and \( B(3, 3) \), we need to first find the correct vector equation of the line passing through these points.

1. **Find the direction vector \(\mathbf{d}\) of the line:**
   \[
   \mathbf{d} = B - A = (3 - (-1), 3 - (-6)) = (4, 9)
   \]

2. **Vector equation of the line:**
   The vector equation of a line passing through point \( A \) with direction vector \(\mathbf{d}\) is:
   \[
   \mathbf{r} = \mathbf{r_0} + t\mathbf{d}
   \]
   where \(\mathbf{r_0}\) is the position vector of point \( A \) and \( t \) is a scalar parameter.

   Substituting \(\mathbf{r_0} = (-1, -6)\) and \(\mathbf{d} = (4, 9)\):
   \[
   \mathbf{r} = (-1, -6) + t(4, 9)
   \]

3. **Possible forms of vector equations:**
   \[
   \mathbf{r} = (-1, -6) + t(4, 9)
   \]
   or equivalently:
   \[
   \mathbf{r} = (3, 3) + t(4, 9)
   \]
   because \( B \) is also on the line, and direction vector remains the same.

4. **Check given vector equations:**
   Let's consider different vector equations to determine which one does not match the above forms.

Please provide the given vector equations, and I'll identify which one does not represent the line passing through points \( A \) and \( B \).

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In the meantime, here's an example of how you could be asked to identify incorrect equations:

1. \(\mathbf{r} = (-1, -6) + t(4, 9)\)
2. \(\mathbf{r} = (3, 3) + t(4, 9)\)
3. \(\mathbf{r} = (-1, -6) + t(3, 7)\)
4. \(\mathbf{r} = (3, 3) + t(4, -9)\)

Given these options, the third and fourth equations are incorrect because their direction vectors do not match the calculated \((4, 9)\).

Would you like to proceed with more details or have specific vector equations for review?

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### Relative Questions:

1. How do you find the direction vector between two points?
2. What is the general form of a vector equation of a line?
3. How can you verify if a point lies on a given line?
4. How do you convert a vector equation into parametric form?
5. What are the differences between parametric and symmetric forms of a line?
6. How do you find the scalar parameter for a specific point on the line?
7. Can a line be uniquely determined by its direction vector and one point?
8. What methods can be used to find the intersection of two lines in vector form?

### Tip:
When verifying vector equations of a line, ensure the direction vectors are consistent and the initial points align with the given points.

Learn how to identify the vector equation that does not represent the line passing through points A(-1, -6) and B(3, 3). This problem covers vectors, direction vector calculation, and vector equations in 2D geometry. Suitable for students in Grades 10-12.

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