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.