Step 1: Understanding the Concept:
In three-dimensional space, a line is uniquely determined by a point through which it passes and its direction.
The direction is often given as a vector. Any line parallel to a vector \( \vec{v} \) will have the same direction ratios as the components of that vector.
The symmetric form of a line's equation is a standard way to represent this relationship.
Step 2: Key Formula or Approach:
The equation of a line passing through the point \( (x_1, y_1, z_1) \) with direction ratios \( (a, b, c) \) is:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
Step 3: Detailed Explanation:
Step 1: Identify the given point \( (x_1, y_1, z_1) \).
The point is \( (1, 2, 3) \).
So, \( x_1 = 1, y_1 = 2, z_1 = 3 \).
Step 2: Identify the direction ratios \( (a, b, c) \).
The line is parallel to the vector \( \vec{v} = 1\hat{i} + 2\hat{j} + 3\hat{k} \).
The coefficients of \( \hat{i}, \hat{j}, \hat{k} \) are the direction ratios.
So, \( a = 1, b = 2, c = 3 \).
Step 3: Substitute these values into the standard symmetric equation form.
\[ \frac{x - (1)}{1} = \frac{y - (2)}{2} = \frac{z - (3)}{3} \]
This simplifies to:
\[ \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} \]
Step 4: Final Answer:
The equation of the line is \( \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3} \).
This corresponds to Option (A).