https://file.aiquickdraw.com/mathbot/file/1724167666519wmnje1sc.jpg

from PIL import Image

# Load the image to examine question 6
image_path = "/mnt/data/1724167666519wmnje1sc.jpg"
image = Image.open(image_path)
image.show()It looks like I have displayed the image containing question 6. To proceed, I’ll need to extract the content of that question. Let me examine it more closely.Question 6 reads:

**"Find parametric equations of the line that passes through the points \( A(3, 1, 2) \) and \( B(5, 4, -6) \)."**

### Solution:

1. **Direction Vector:**
   The direction vector \(\vec{d}\) of the line can be found by subtracting the coordinates of point \(A\) from point \(B\):

   \[
   \vec{d} = B - A = (5 - 3, 4 - 1, -6 - 2) = (2, 3, -8)
   \]

2. **Parametric Equations:**
   The parametric equations of a line passing through a point \( A(x_1, y_1, z_1) \) and having a direction vector \(\vec{d} = (d_x, d_y, d_z)\) are given by:

   \[
   x = x_1 + t \cdot d_x
   \]
   \[
   y = y_1 + t \cdot d_y
   \]
   \[
   z = z_1 + t \cdot d_z
   \]
   where \(t\) is a parameter.

   Using \(A(3, 1, 2)\) as the point on the line and \(\vec{d} = (2, 3, -8)\), the parametric equations are:

   \[
   x = 3 + 2t
   \]
   \[
   y = 1 + 3t
   \]
   \[
   z = 2 - 8t
   \]

Thus, the parametric equations of the line passing through \( A(3, 1, 2) \) and \( B(5, 4, -6) \) are:
\[
x = 3 + 2t, \quad y = 1 + 3t, \quad z = 2 - 8t
\]

Would you like further details on this solution? Here are five questions to help you deepen your understanding:

1. What are the vector equations of a line, and how do they relate to parametric equations?
2. How do you calculate the direction vector between two points in three-dimensional space?
3. How do you convert parametric equations into symmetric or vector form?
4. What is the significance of the parameter \(t\) in parametric equations?
5. How can parametric equations be used to find the intersection of two lines?

**Tip:** Parametric equations are especially useful in 3D geometry because they represent the coordinates of points along a line based on a variable parameter.

Learn how to find parametric equations of a line passing through two points in 3D space. This problem involves calculating direction vectors, parametric equations, and understanding the significance of parameters in 3D geometry. Suitable for students in Grades 10-12.

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