To find the parametric equation of a line passing through two points $A(2, 5, -4)$ and $B(-2, 2, -6)$, we can express the line using the parameter $t$.

Step 1: Find the direction vector

The direction vector $\mathbf{v}$ is found by subtracting the coordinates of point $A$ from point $B$:

$\mathbf{v} = B - A = (-2 - 2, 2 - 5, -6 + 4) = (-4, -3, -2)$

Step 2: Parametric equation of the line

The parametric equation of the line is given by:

$\mathbf{r}(t) = \mathbf{A} + t \cdot \mathbf{v}$

Where:

• $\mathbf{A} = (2, 5, -4)$ is the position vector of point $A$,
• $\mathbf{v} = (-4, -3, -2)$ is the direction vector,
• $t$ is the parameter.

Thus, the parametric equations for $x(t)$, $y(t)$, and $z(t)$ are:

$x(t) = 2 - 4t$ $y(t) = 5 - 3t$ $z(t) = -4 - 2t$

Final Parametric Equation

The parametric equation of the line is:

$\mathbf{r}(t) = \langle 2 - 4t, 5 - 3t, -4 - 2t \rangle$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would you find the parametric equations if the points were given in 2D instead of 3D?
2. Can you convert this parametric equation into its symmetric form?
3. What is the relationship between the parameter $t$ and the points on the line?
4. How would you find the equation of a plane containing this line and another point?
5. How do you determine if two parametric equations represent the same line?

Tip: The direction vector of the line is crucial—it determines the orientation and slope of the line in space.