equation of The line passing through (−5, 0, 4) and (6, −3, 2)

To find the equation of the line passing through two points in 3D space, we need to use the **vector form** of the line equation. The general equation of a line passing through two points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) can be written as:

\[
\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{d}
\]

where:
- \( \mathbf{r}(t) \) is the position vector of a point on the line,
- \( \mathbf{r}_0 \) is the position vector of the first point \( P_1 \),
- \( \mathbf{d} \) is the direction vector (calculated by subtracting the coordinates of \( P_2 \) and \( P_1 \)),
- \( t \) is a scalar parameter.

### Step 1: Find the direction vector
The direction vector \( \mathbf{d} \) is given by:

\[
\mathbf{d} = P_2 - P_1 = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
\]

Substitute the coordinates of the points \( P_1(-5, 0, 4) \) and \( P_2(6, -3, 2) \):

\[
\mathbf{d} = (6 - (-5), -3 - 0, 2 - 4) = (11, -3, -2)
\]

### Step 2: Write the parametric equation of the line
Using the direction vector \( \mathbf{d} = (11, -3, -2) \) and the point \( P_1(-5, 0, 4) \), the parametric equations of the line are:

\[
x = -5 + 11t
\]
\[
y = 0 - 3t
\]
\[
z = 4 - 2t
\]

### Final answer: 
The parametric equations of the line passing through the points \( (-5, 0, 4) \) and \( (6, -3, 2) \) are:

\[
\boxed{x = -5 + 11t, \, y = -3t, \, z = 4 - 2t}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you find the equation of a plane given three points?
2. What is the parametric form of a line in 2D space?
3. How do you convert a parametric equation into a vector equation?
4. How do you determine if two lines in 3D space are parallel?
5. What is the distance between two points in 3D space?

**Tip:** The direction vector gives important information about the orientation of the line, such as its slope and direction.

equation of the line passing through (−5, 0, 4) and (6, −3, 2)

Learn how to find the parametric equation of a line in 3D space passing through points (-5, 0, 4) and (6, -3, 2). This solution covers key 3D geometry concepts, including direction vectors and parametric equations. Suitable for students in Grades 11-12.

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