A line passing through the points A (-2 , -1 , 5) and B ( 1, 3 , -1) ,find the equation of the line in parametric form. Also, write the equation in non - parametric form.

Answer

To find the equation of the line passing through the points A (-2, -1, 5) and B (1, 3, -1) in both parametric and non-parametric forms, we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point A: \( A(-2, -1, 5) \)
- Point B: \( B(1, 3, -1) \)

### Step 2: Find the Direction Vector
The direction vector \( \mathbf{b} \) of the line can be found by subtracting the coordinates of point A from point B:
\[ \mathbf{b} = B - A = (1 - (-2), 3 - (-1), -1 - 5) = (3, 4, -6) \]

### Step 3: Write the Position Vector of Point A
The position vector \( \mathbf{a} \) of point A can be expressed in vector form:
\[ \mathbf{a} = -2\mathbf{i} - 1\mathbf{j} + 5\mathbf{k} \]

### Step 4: Write the Parametric Form of the Line
The parametric form of the line can be written as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
Substituting the values we found:
\[ \mathbf{r} = (-2\mathbf{i} - 1\mathbf{j} + 5\mathbf{k}) + \lambda(3\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}) \]
This can be expressed in terms of its components:
\[ \mathbf{r} = (-2 + 3\lambda)\mathbf{i} + (-1 + 4\lambda)\mathbf{j} + (5 - 6\lambda)\mathbf{k} \]

### Step 5: Write the Non-Parametric Form of the Line
The non-parametric form of the line can be expressed using the direction ratios derived from the direction vector \( \mathbf{b} \):
\[ \frac{x - (-2)}{3} = \frac{y - (-1)}{4} = \frac{z - 5}{-6} \]
This simplifies to:
\[ \frac{x + 2}{3} = \frac{y + 1}{4} = \frac{z - 5}{-6} \]

### Summary of the Solutions
- **Parametric Form**:
\[ \mathbf{r} = (-2 + 3\lambda)\mathbf{i} + (-1 + 4\lambda)\mathbf{j} + (5 - 6\lambda)\mathbf{k} \]
- **Non-Parametric Form**:
\[ \frac{x + 2}{3} = \frac{y + 1}{4} = \frac{z - 5}{-6} \]