To find the vector equation and the parametric equations of the line, let's break it down step by step.

Step 1: Vector Equation of the Line

The vector equation of a line is given by the formula:

$\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}$

Where:

• $\mathbf{r}_0$ is the position vector of a point on the line (we are given the point $(8, -9, 4)$).
• $\mathbf{v}$ is the direction vector of the line (we are given the vector $\langle 1, 5, -2 \rangle$).
• $t$ is a parameter.

So, the vector equation becomes:

$\mathbf{r}(t) = \langle 8, -9, 4 \rangle + t \langle 1, 5, -2 \rangle$

Simplifying this expression:

$\mathbf{r}(t) = \langle 8 + t, -9 + 5t, 4 - 2t \rangle$

Thus, the vector equation is:

$\mathbf{r}(t) = \langle 8 + t, -9 + 5t, 4 - 2t \rangle$

Step 2: Parametric Equations of the Line

To get the parametric equations, we simply express each component of the vector equation in terms of $t$. The parametric equations are:

$x(t) = 8 + t$ $y(t) = -9 + 5t$ $z(t) = 4 - 2t$

Final Answer

• The vector equation is:

$\mathbf{r}(t) = \langle 8 + t, -9 + 5t, 4 - 2t \rangle$

• The parametric equations are:

$x(t) = 8 + t, \quad y(t) = -9 + 5t, \quad z(t) = 4 - 2t$

Would you like a further breakdown of these steps, or is there anything else you'd like to discuss?

Here are a few related questions:

1. How would the parametric equations change if the line were in 2D instead of 3D?
2. What is the geometric interpretation of a vector equation of a line?
3. How would you find the direction vector if only two points on the line are given?
4. How do you convert a vector equation of a line into parametric form?
5. What is the relationship between the direction vector and the slope of a line in 2D?

Tip: Always verify the direction vector by ensuring it points in the correct direction and check the point for consistency with the line.