To complete the parametric equations for the line through the points $(-2, 0, -8)$ and $(5, 7, 8)$, we need to use the following general form for a parametric line:

$x(t) = x_0 + at, \quad y(t) = y_0 + bt, \quad z(t) = z_0 + ct$

where:

• $(x_0, y_0, z_0)$ is a point on the line,
• $(a, b, c)$ is the direction vector of the line, obtained by subtracting the two points.

Step 1: Find the Direction Vector

The direction vector is obtained by subtracting the coordinates of the two points: $(5 - (-2), 7 - 0, 8 - (-8)) = (7, 7, 16)$

So, the direction vector is $(7, 7, 16)$.

Step 2: Write Parametric Equations

We already have the $x(t)$ equation: $x(t) = -2 + 7t$

Using the point $(-2, 0, -8)$ and the direction vector $(7, 7, 16)$, the parametric equations are: $y(t) = 0 + 7t = 7t$ $z(t) = -8 + 16t$

Final Answer

The complete parametric equations for the line are: $x(t) = -2 + 7t$ $y(t) = 7t$ $z(t) = -8 + 16t$

Would you like more details on the solution or have any questions?

Related Questions

1. How do you find the parametric equation for a line in 3D space?
2. What is the significance of the direction vector in parametric equations?
3. How can parametric equations be used to find intersection points between lines?
4. What would change if we used a different point on the line as a starting point?
5. How would we convert this parametric line equation into symmetric form?

Tip

To find parametric equations for a line through two points, subtract the coordinates to get the direction vector and use either point as the starting position.