Express the following equation of the lines into vector form :
`(x - 3)/(3) = (y - 8)/(-1) = (z - 3)/(1)`
and `(x + 3)/(-3) = (y + 7)/(2) = (z - 6)/(4)`

Answer

To express the given equations of the lines in vector form, we will follow these steps:

### Step 1: Identify the components of the first line
The first line is given by the equation:
\[ \frac{x - 3}{3} = \frac{y - 8}{-1} = \frac{z - 3}{1} \]
From this equation, we can identify the following:
- The point through which the line passes (A, B, C) is (3, 8, 3).
- The direction ratios (L, M, N) are (3, -1, 1).

### Step 2: Write the vector form of the first line
Using the standard vector form of a line:
\[ \vec{r} = \vec{a} + \lambda \vec{d} \]
where \(\vec{a}\) is the position vector of the point (3, 8, 3) and \(\vec{d}\) is the direction vector (3, -1, 1), we can write:
\[ \vec{r_1} = (3\hat{i} + 8\hat{j} + 3\hat{k}) + \lambda (3\hat{i} - \hat{j} + \hat{k}) \]
This simplifies to:
\[ \vec{r_1} = (3 + 3\lambda)\hat{i} + (8 - \lambda)\hat{j} + (3 + \lambda)\hat{k} \]

### Step 3: Identify the components of the second line
The second line is given by the equation:
\[ \frac{x + 3}{-3} = \frac{y + 7}{2} = \frac{z - 6}{4} \]
From this equation, we can identify:
- The point through which the line passes (A, B, C) is (-3, -7, 6).
- The direction ratios (L, M, N) are (-3, 2, 4).

### Step 4: Write the vector form of the second line
Using the same vector form:
\[ \vec{r} = \vec{a} + \mu \vec{d} \]
where \(\vec{a}\) is the position vector of the point (-3, -7, 6) and \(\vec{d}\) is the direction vector (-3, 2, 4), we can write:
\[ \vec{r_2} = (-3\hat{i} - 7\hat{j} + 6\hat{k}) + \mu (-3\hat{i} + 2\hat{j} + 4\hat{k}) \]
This simplifies to:
\[ \vec{r_2} = (-3 - 3\mu)\hat{i} + (-7 + 2\mu)\hat{j} + (6 + 4\mu)\hat{k} \]

### Final Answer
The vector forms of the lines are:
1. For the first line:
\[ \vec{r_1} = (3 + 3\lambda)\hat{i} + (8 - \lambda)\hat{j} + (3 + \lambda)\hat{k} \]
2. For the second line:
\[ \vec{r_2} = (-3 - 3\mu)\hat{i} + (-7 + 2\mu)\hat{j} + (6 + 4\mu)\hat{k} \]