(b) Find parametric equations of the line that passes through the point 
(1, −4)
 and is perpendicular to the line with vector equation 
r(t) = 

−8 + 4t, 1 + 3t

. (Enter your answer as a comma-separated list of equations where x and y are in terms of the parameter t.)

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?

---

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.

Find parametric equations of the line that passes through the point (1, −4) and is perpendicular to the line with vector equation r(t) = −8 + 4t, 1 + 3t.

Learn how to find the parametric equations of a line passing through the point (1, -4) and perpendicular to the line with vector equation r(t) = −8 + 4t, 1 + 3t. This problem involves key concepts from analytic geometry, including vector equations and the dot product condition for perpendicularity. Suitable for high school students in Grades 10-12.

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