$$ 1 \leftarrow $$ Find an equation of the line passing through $$ P_{0} $$ and normal to the plane $$ P $$. $$ P_{0}(1,2,3) ; P: 2 x-3 y+z=10 $$ Choose the correct answer below. A. An equation of the line is $$ \langle 1-2 t, 2+3 t, 3-t\rangle $$. B. An equation of the line is $$ \langle 2-t,-3-2 t, 1-3 t\rangle $$. An equation of the line is $$ \langle 2-t,-3-2 t, 1-3 t\rangle $$. D. An equation of the line is $$ \langle 1+2 t, 2-3 t, 3+t\rangle $$.

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To find an equation of the line passing through the point $$ P_0(1, 2, 3) $$ and normal to the plane $$ P: 2x - 3y + z = 10 $$, follow these steps:

1. **Identify the Normal Vector of the Plane:**
   The normal vector ($$ \mathbf{n} $$) of the plane $$ P $$ is derived from the coefficients of $$ x $$, $$ y $$, and $$ z $$ in the plane equation:
   $$
   \mathbf{n} = \langle 2, -3, 1 \rangle
   $$

2. **Use the Point and Direction Vector to Formulate the Line:**
   A line passing through $$ P_0(1, 2, 3) $$ with direction vector $$ \mathbf{n} = \langle 2, -3, 1 \rangle $$ can be expressed parametrically as:
   $$
   \begin{cases}
   x = 1 + 2t \
   y = 2 - 3t \
   z = 3 + t
   \end{cases}
   $$
   This can also be written in vector form as:
   $$
   \langle 1 + 2t, 2 - 3t, 3 + t \rangle
   $$

3. **Match with the Given Options:**
   Comparing with the provided options, the correct parametric equation of the line is:

**D. An equation of the line is $$ \langle 1+2t, 2-3t, 3+t \rangle $$.**

**Answer:**  
**D. An equation of the line is $$ \langle 1+2 t, 2-3 t, 3+t\rangle $$.**
The correct equation of the line is $$ \langle 1+2t, 2-3t, 3+t \rangle $$.

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