Find a vector equation and parametric equations for the line segment that joins P to Q.
P(1, 0, 1), Q(5, 8, 1)

(a) Vector equation.
<b>r</b>(t) = langle1+4t, 
  , 
  rangle

(b) Parametric equations.
x(t) = 1+4t
y(t) = 
 
z(t) = 
 

To find the vector and parametric equations for the line segment joining points \( P(1, 0, 1) \) and \( Q(5, 8, 1) \), follow these steps:

### Step 1: Vector equation

The vector equation of a line is given by:

\[
\mathbf{r}(t) = \mathbf{P} + t(\mathbf{Q} - \mathbf{P})
\]

where:
- \( \mathbf{P} \) is the position vector of point \( P(1, 0, 1) \),
- \( \mathbf{Q} \) is the position vector of point \( Q(5, 8, 1) \),
- \( t \) is a parameter that varies from 0 to 1 for the line segment.

First, calculate \( \mathbf{Q} - \mathbf{P} \):
\[
\mathbf{Q} - \mathbf{P} = (5 - 1, 8 - 0, 1 - 1) = (4, 8, 0)
\]

So, the vector equation becomes:
\[
\mathbf{r}(t) = \langle 1, 0, 1 \rangle + t \langle 4, 8, 0 \rangle
\]

Expanding the vector equation:

\[
\mathbf{r}(t) = \langle 1 + 4t, 0 + 8t, 1 + 0t \rangle
\]

Thus, the vector equation is:

\[
\mathbf{r}(t) = \langle 1 + 4t, 8t, 1 \rangle
\]

### Step 2: Parametric equations

From the vector equation \( \mathbf{r}(t) = \langle 1 + 4t, 8t, 1 \rangle \), we can write the parametric equations as follows:

\[
x(t) = 1 + 4t
\]
\[
y(t) = 8t
\]
\[
z(t) = 1
\]

These parametric equations describe the line segment between points \( P \) and \( Q \).

### Summary:

- **Vector equation:**

  \[
  \mathbf{r}(t) = \langle 1 + 4t, 8t, 1 \rangle
  \]

- **Parametric equations:**

  \[
  x(t) = 1 + 4t, \quad y(t) = 8t, \quad z(t) = 1
  \]

Would you like more details on this process, or do you have any questions?

### 5 Related Questions:
1. How do you find the direction vector between two points in 3D space?
2. What is the geometric meaning of parametric equations for a line?
3. How does the parameter \( t \) control the line segment in parametric equations?
4. How do vector equations differ when representing lines vs. line segments?
5. What happens to the parametric equations if points \( P \) and \( Q \) lie on different planes?

### Tip:
In parametric equations, \( t = 0 \) corresponds to point \( P \), and \( t = 1 \) corresponds to point \( Q \), which helps limit the line to the segment between these points.

Find a vector equation and parametric equations for the line segment that joins P to Q. P(1, 0, 1), Q(5, 8, 1)

Learn how to find the vector equation and parametric equations for the line segment joining points P(1, 0, 1) and Q(5, 8, 1). This problem involves 3D geometry concepts, vector equations, and parametric forms suitable for students in Grades 10-12.

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