To write the parametric equations of the line passing through the point $(6, 0, -4)$ and parallel to the vector $\langle 4, 1, -6 \rangle$, we follow the general formula for parametric equations of a line:

$\text{Point on the line: } (x_0, y_0, z_0)$ $\text{Direction vector: } \langle a, b, c \rangle$ The parametric equations are given by:

$x = x_0 + at$ $y = y_0 + bt$ $z = z_0 + ct$

Given:

• Point $(x_0, y_0, z_0) = (6, 0, -4)$
• Direction vector $\langle a, b, c \rangle = \langle 4, 1, -6 \rangle$

Parametric Equations:

$x = 6 + 4t$ $y = 0 + 1t \quad \text{or} \quad y = t$ $z = -4 - 6t$

Thus, the parametric equations for the line are: $x = 6 + 4t, \quad y = t, \quad z = -4 - 6t$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do we find parametric equations if the line is perpendicular to a vector?
2. What are the parametric equations of a line through two given points?
3. How can we convert parametric equations into symmetric equations of a line?
4. How do you determine if two parametric lines are parallel or coincident?
5. How do we find the point of intersection of two parametric lines?

Tip: Parametric equations are useful in describing lines in 3D space, especially for motion and vector analysis.