equation of The line passing through (−1, −1, −1) and (1, −1, 2)

To find the equation of the line passing through the points \((-1, -1, -1)\) and \((1, -1, 2)\) in 3-dimensional space, we use the parametric form of the line equation. The parametric equation of a line that passes through a point \( P_0(x_0, y_0, z_0) \) and has a direction vector \( \vec{d} = (a, b, c) \) is given by:

\[
\begin{aligned}
x &= x_0 + at, \\
y &= y_0 + bt, \\
z &= z_0 + ct,
\end{aligned}
\]

where \( t \) is the parameter.

### Steps to Find the Equation:

1. **Find the direction vector**:
   The direction vector \( \vec{d} \) is obtained by subtracting the coordinates of the two points:
   \[
   \vec{d} = (1 - (-1), -1 - (-1), 2 - (-1)) = (2, 0, 3).
   \]

2. **Write the parametric equations**:
   Choose one of the points, say \( (-1, -1, -1) \), as the initial point \( P_0 \). The parametric equations of the line are:
   \[
   \begin{aligned}
   x &= -1 + 2t, \\
   y &= -1 + 0t, \\
   z &= -1 + 3t.
   \end{aligned}
   \]

   Simplifying these, we get:
   \[
   \begin{aligned}
   x &= -1 + 2t, \\
   y &= -1, \\
   z &= -1 + 3t.
   \end{aligned}
   \]

Therefore, the parametric equation of the line passing through the points \( (-1, -1, -1) \) and \( (1, -1, 2) \) is:
\[
(x, y, z) = (-1 + 2t, -1, -1 + 3t), \quad t \in \mathbb{R}.
\]

### Vector Form:
Alternatively, the vector equation of the line can be expressed as:
\[
\vec{r} = \vec{r_0} + t\vec{d},
\]
where \( \vec{r_0} = (-1, -1, -1) \) and \( \vec{d} = (2, 0, 3) \):
\[
\vec{r} = (-1, -1, -1) + t(2, 0, 3).
\]

Would you like further details or have any questions about this?

Here are 5 related questions for further exploration:

1. How would you find the line equation in vector form given two points?
2. What is the significance of the direction vector in the line equation?
3. How can you find the intersection of two lines in 3D space?
4. How do you determine if two lines in 3D space are parallel or perpendicular?
5. How can you convert the

Find the equation of the line passing through (-1, -1, -1) and (1, -1, 2)

Find the parametric and vector form of the line passing through points (-1, -1, -1) and (1, -1, 2) in 3D space. Learn about direction vectors, parametric equations, and their application in 3D geometry. Suitable for Grades 10-12 students.

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