The cartesian equation of a line is `(x+2)/(1) = (y+3)/(-2) = (z)/(3)`, find its vector equation.

Answer

To convert the given Cartesian equation of the line into its vector equation, we can follow these steps:

### Step 1: Understand the Given Equation
The given Cartesian equation of the line is:
\[ \frac{x + 2}{1} = \frac{y + 3}{-2} = \frac{z}{3} \]

### Step 2: Identify the Components
We can rewrite the equation in the standard form:
\[ \frac{x - (-2)}{1} = \frac{y - (-3)}{-2} = \frac{z - 0}{3} \]
From this, we can identify:
- The point on the line (x₁, y₁, z₁) = (-2, -3, 0)
- The direction ratios (b₁, b₂, b₃) = (1, -2, 3)

### Step 3: Write the Position Vector (a vector)
The position vector \( \mathbf{a} \) corresponding to the point (-2, -3, 0) can be expressed as:
\[ \mathbf{a} = -2 \mathbf{i} - 3 \mathbf{j} + 0 \mathbf{k} = -2 \mathbf{i} - 3 \mathbf{j} \]

### Step 4: Write the Direction Vector (b vector)
The direction vector \( \mathbf{b} \) corresponding to the direction ratios (1, -2, 3) can be expressed as:
\[ \mathbf{b} = 1 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k} \]

### Step 5: Write the Vector Equation of the Line
The vector equation of the line can be written as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
Substituting the values of \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{r} = (-2 \mathbf{i} - 3 \mathbf{j}) + \lambda (1 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k}) \]

### Final Vector Equation
Thus, the vector equation of the line is:
\[ \mathbf{r} = (-2 + \lambda) \mathbf{i} + (-3 - 2\lambda) \mathbf{j} + (3\lambda) \mathbf{k} \]