Write the equation of a line , parallel to the line `(x - 2)/(-3) = (y + 3)/(2) = (z + 3)` and passing through the point (1,2,3).

Answer

To find the equation of a line that is parallel to the given line \((x - 2)/(-3) = (y + 3)/(2) = (z + 3)\) and passes through the point \((1, 2, 3)\), we can follow these steps:

### Step 1: Identify the direction ratios of the given line
The given line can be expressed in the symmetric form:
\[ \frac{x - 2}{-3} = \frac{y + 3}{2} = \frac{z + 3}{1} \]
From this, we can extract the direction ratios of the line, which are \((-3, 2, 1)\).

### Step 2: Write the point through which the new line passes
The new line must pass through the point \((1, 2, 3)\). We denote this point as \(A\), so:
\[ A = (1, 2, 3) \]

### Step 3: Write the equation of the line
The equation of a line in vector form can be expressed as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
where \(\mathbf{a}\) is a position vector of a point on the line, \(\lambda\) is a scalar parameter, and \(\mathbf{b}\) is the direction vector of the line.

Here, we have:
- \(\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) (the position vector of point A)
- \(\mathbf{b} = \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix}\) (the direction ratios we found)

Thus, the equation of the line can be written as:
\[ \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix} \]

### Step 4: Write the parametric equations
From the vector equation, we can derive the parametric equations:
\[ x = 1 - 3\lambda \]
\[ y = 2 + 2\lambda \]
\[ z = 3 + \lambda \]

### Final Answer
The equation of the line parallel to the given line and passing through the point \((1, 2, 3)\) is:
\[ \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix} \]
or in parametric form:
\[ x = 1 - 3\lambda, \quad y = 2 + 2\lambda, \quad z = 3 + \lambda \]