The equation of the line in vector form passing through the point `(-1,3,5)` and parallel to the line `(x-3)/(2) = ( y-4)/(3) , z=2` is

Options

A. `overset(to)( r) = - hat(i) +3 hat(j) + 5 hat(k) + lambda (2 hat(i) + 3 hat(j) + hat(k) )`
B. `overset(to)( r) = - hat(i) + 3 hat(j) + 5 hat(k) + lambda (2 hat(i) + 3 hat(i) )`
C. `overset(to)( r) =2 hat(i) +3 hat(j) - 2 hat(k) + lambda (- hat(i) +3 hat(j) + 5 hat(k) )`
D. `overset(to)( r) = 2 hat(i) + 3 hat(j) + lambda (- hat(i) + 3 hat(j) +5 hat(k) )`

Answer

To find the equation of the line in vector form that passes through the point \((-1, 3, 5)\) and is parallel to the given line, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Line**:
The given line is represented as:
\[ \frac{x - 3}{2} = \frac{y - 4}{3} = z - 2 \]
This can be interpreted as a parametric equation of the line.

2. **Extract Direction Ratios**:
From the equation, we can rewrite it in the form:
\[ \frac{x - 3}{2} = \frac{y - 4}{3} = \frac{z - 2}{0} \]
Here, the direction ratios of the line can be taken as \(2, 3, 0\).

3. **Determine the Point and Direction Vector**:
The point through which our new line passes is given as \((-1, 3, 5)\), which can be represented as a position vector:
\[ \mathbf{a} = -1 \mathbf{i} + 3 \mathbf{j} + 5 \mathbf{k} \]
The direction vector \(\mathbf{b}\) corresponding to the direction ratios \(2, 3, 0\) is:
\[ \mathbf{b} = 2 \mathbf{i} + 3 \mathbf{j} + 0 \mathbf{k} \]

4. **Write the Vector Equation of the Line**:
The vector equation of a line can be expressed as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
Substituting the values of \(\mathbf{a}\) and \(\mathbf{b}\):
\[ \mathbf{r} = (-1 \mathbf{i} + 3 \mathbf{j} + 5 \mathbf{k}) + \lambda (2 \mathbf{i} + 3 \mathbf{j} + 0 \mathbf{k}) \]

5. **Simplify the Equation**:
This can be simplified to:
\[ \mathbf{r} = (-1 + 2\lambda) \mathbf{i} + (3 + 3\lambda) \mathbf{j} + 5 \mathbf{k} \]

### Final Vector Equation:
Thus, the vector equation of the line is:
\[ \mathbf{r} = (-1 + 2\lambda) \mathbf{i} + (3 + 3\lambda) \mathbf{j} + 5 \mathbf{k} \]