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.