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The given problem presents the parametric equations of a line \( r \) and asks to find the vector equation of a line parallel to \( r \) that passes through the point \( B(1, 2, 1) \).

### Step 1: Identify the direction vector of line \( r \)
The parametric form of the line \( r \) is:
\[
\begin{aligned}
x &= 2 + 3t \\
y &= 1 + t \\
z &= 5 + 4t
\end{aligned}
\]
From these equations, we can extract the direction vector of line \( r \) by looking at the coefficients of \( t \). Thus, the direction vector of line \( r \) is:
\[
\mathbf{v} = \langle 3, 1, 4 \rangle
\]

### Step 2: Equation of the parallel line
A line that is parallel to \( r \) will have the same direction vector \( \mathbf{v} = \langle 3, 1, 4 \rangle \), but it will pass through a different point, specifically \( B(1, 2, 1) \).

The vector equation of a line is given by:
\[
\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}
\]
Where:
- \( \mathbf{r}_0 = \langle 1, 2, 1 \rangle \) is the position vector of point \( B \),
- \( \mathbf{v} = \langle 3, 1, 4 \rangle \) is the direction vector,
- \( t \in \mathbb{R} \) is the parameter.

Thus, the parametric equations of the line parallel to \( r \) and passing through \( B(1, 2, 1) \) are:
\[
\begin{aligned}
x &= 1 + 3t \\
y &= 2 + t \\
z &= 1 + 4t
\end{aligned}
\]

This is the equation of the required line.

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Would you like more details, or do you have any further questions?

Here are five related questions you could explore:
1. How do you determine if two lines are parallel in three dimensions?
2. What is the general form of a line in vector form in three dimensions?
3. Can two parallel lines intersect? If so, under what conditions?
4. How do you find the shortest distance between two parallel lines?
5. What changes in the equation if the lines are skew (non-parallel)?

**Tip:** To find if two lines are parallel, check if their direction vectors are scalar multiples of each other.

Consider the line r with parametric equations: x = 2 + 3t, y = 1 + t, z = 5 + 4t. Choose the vector equation of the line parallel to r passing through point B(1, 2, 1).

Learn how to find the vector equation of a line parallel to a given line with parametric equations and passing through a specific point in 3D space. This problem explores key concepts such as parametric equations, direction vectors, and vector equations of lines in 3D geometry.

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