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To find the general equation of a line in \(\mathbb{R}^2\) passing through the point \((1, 3)\) and parallel to the vector \(\begin{pmatrix} 2 \\ -7 \end{pmatrix}\), let's proceed as follows:

### Step 1: Identify the Slope of the Line
The direction vector of the line is given by \(\begin{pmatrix} 2 \\ -7 \end{pmatrix}\). A vector \(\begin{pmatrix} a \\ b \end{pmatrix}\) gives the slope of the line as \(\frac{b}{a}\). Therefore, the slope \(m\) of our line is:
\[
m = \frac{-7}{2}
\]

### Step 2: Use the Point-Slope Form
The point-slope form of a line passing through \((x_1, y_1)\) with slope \(m\) is:
\[
y - y_1 = m(x - x_1)
\]
Using \((x_1, y_1) = (1, 3)\) and \(m = -\frac{7}{2}\), we get:
\[
y - 3 = -\frac{7}{2}(x - 1)
\]

### Step 3: Simplify to Standard Form
Now, expand and rearrange the equation into a standard form (Ax + By = C):
\[
y - 3 = -\frac{7}{2}x + \frac{7}{2}
\]
Multiply through by 2 to eliminate the fraction:
\[
2y - 6 = -7x + 7
\]
Rearrange terms to get:
\[
7x + 2y = 13
\]

### Conclusion
The equation of the line is:
\[
7x + 2y = 13
\]
This matches option **A** in the provided choices.

### Answer:
**The correct answer is A: \(7x + 2y = 13\)**.

Would you like more details, or have any questions?

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Here are 5 related questions to expand on this topic:

1. How would you find the equation of a line perpendicular to a given vector?
2. What is the general form of a line equation in \(\mathbb{R}^2\)?
3. How can you determine if two lines are parallel or perpendicular?
4. What would the line equation be if it passed through a different point with the same direction vector?
5. How can vector notation help in representing lines in higher dimensions?

**Tip:** To find a line parallel to a vector, the direction vector of the line and the given vector should be the same, meaning they share the same slope or direction ratio.

Find the general equation of a line in \(\mathbb{R}^2\) passing through the point \((1, 3)\) and parallel to \(\begin{pmatrix} 2 \\ -7 \end{pmatrix}\).

Learn how to find the equation of a line in \(\mathbb{R}^2\) that passes through the point \((1, 3)\) and is parallel to the vector \(\begin{pmatrix} 2 \\ -7 \end{pmatrix}\). This solution covers concepts in coordinate geometry and the use of point-slope form for lines. Suitable for students in Grades 10-12.

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