The question aims to find the **vector equation** and the **parametric equations** for the line that joins two points, **P and Q.** The points **P and Q are given.**

The question depends on the concepts of the **vector equation** of the **line.** The **vector equation** for a **finite line** with $r_0$ as the **initial point** of the line. The **parametric equation** of **two vectors** joined by a **finite line** is given as:

\[ r(t) = (1\ -\ t) r_0 + tr_1 \hspace{0.2in} where \hspace{0.2in} 0 \leq t \leq 1 \]

## Expert Answer

The vectors** P and Q** are given as:

\[ P = < -1, 0, 1 > \]

\[ Q = < -2.5, 0, 2.1 > \]

Here, taking **P** as the first vector as $r_0$ and **Q** as the second vector as$r_1$.

Substituting the values of both **vectors** in the **parametric equation,** we get:

\[ r(t) = ( 1\ -\ t) < -1, 0, 1 > + t < -2.5, 0, 2.1 > \]

\[ r(t) = < -1 + t , 0, 1\ -\ t > + < -2.5t, 0, 2.1t > \]

\[ r(t) = < -1 + t\ -\ 2.5t, 0 + 0, 1\ -\ t + 2.1t > \]

\[ r(t) = < -1\ -\ 1.5t, 0, 1 + 1.1t > \]

The **corresponding parametric equations** of the **line** are calculated to be:

\[ x = -1\ -\ 1.5t \hspace{0.2in} | \hspace{0.2in} y = 0 \hspace{0.2in} | \hspace{0.2in} z = 1 + 1.1t \]

Where the value to t only ranges from [0, 1].

## Numerical Result

The **parametric equation** of the line joining **P and Q** is calculated to be:

\[ r(t) = < -1\ -\ 1.5t, 0, 1 + 1.1t > \]

The corresponding **parametric equations** of the **line** are calculated to be:

\[ x = -1\ -\ 1.5t \hspace{0.2in} | \hspace{0.2in} y = 0 \hspace{0.2in} | \hspace{0.2in} z = 1 + 1.1t \]

Where the value to t only ranges from [0, 1].

## Example

The **vectors** $r_0$ and **v** are given below. Find the **vector equation** of the **line** containing $r_0$ **parallel** to** v.**

\[ r_0 = < -1, 2, -1 > \]

\[ v = < 1, -3, 0 > \]

We can use the **vector equation** of the **line,** which is given as:

\[ r(t) = r_0 + tv \]

Substituting the values, we get:

\[ r(t) = < -1, 2, -1 > + t < 1, -3, 0 > \]

\[ r(t) = < -1, 2, -1 > + < t, -3t, 0 > \]

\[ r(t) = < -1 + t, 2\ -\ 3t, -1 > \]

The corresponding **parametric equations** are calculated to be:

\[ x = 1 + t \hspace{0.2in} | \hspace{0.2in} y = 2\ -\ 3t \hspace{0.2in} | \hspace{0.2in} z = -1 \]