To find the parametric equations of the line that passes through the point $(1, -4)$ and is perpendicular to the line given by the vector equation:

$\mathbf{r}(t) = \langle -8 + 4t, 1 + 3t \rangle$

we can follow these steps:

Step 1: Determine the direction vector of the given line

The direction vector of the line given by $\mathbf{r}(t)$ is:

$\mathbf{v} = \langle 4, 3 \rangle$

Step 2: Find the direction vector of the perpendicular line

Since the new line is perpendicular to the given line, its direction vector $\mathbf{w}$ must satisfy the dot product condition:

$\mathbf{v} \cdot \mathbf{w} = 0$

Let $\mathbf{w} = \langle a, b \rangle$. Then,

$4a + 3b = 0$

This equation shows that $a$ and $b$ are related by:

$b = -\frac{4}{3}a$

We can choose a convenient value for $a$, say $a = 3$, then:

$b = -4$

Thus, the direction vector $\mathbf{w}$ for the perpendicular line is:

$\mathbf{w} = \langle 3, -4 \rangle$

Step 3: Write the parametric equations

The line passes through the point $(1, -4)$ and has the direction vector $\langle 3, -4 \rangle$. The parametric equations for the line are:

$x(t) = 1 + 3t, \quad y(t) = -4 - 4t$

So the final answer is:

$\boxed{x(t) = 1 + 3t, \quad y(t) = -4 - 4t}$

Do you want further details or have any questions?

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1. How do you determine the direction vector of a line from its vector equation?
2. Why is the dot product used to find perpendicular lines?
3. What happens if you choose different values for the parameter $a$ in the direction vector?
4. How can you convert parametric equations to a vector equation?
5. What is the significance of the point $(1, -4)$ in the parametric equations?

Tip: Always check the direction vector of a line before assuming it is perpendicular or parallel to another.