Find the vector equation of the line which passes through the point (3,4,5) and is parallel the vector `2 hat(i) + 2 hat(j) - 3 hat(k)`.

Answer

To find the vector equation of the line that passes through the point \( (3, 4, 5) \) and is parallel to the vector \( \mathbf{b} = 2\hat{i} + 2\hat{j} - 3\hat{k} \), we can follow these steps:

### Step 1: Identify the point and direction vector
The point through which the line passes is given as \( (3, 4, 5) \). We can represent this point as a position vector:
\[ \mathbf{A} = 3\hat{i} + 4\hat{j} + 5\hat{k} \]
The direction vector of the line is given as:
\[ \mathbf{B} = 2\hat{i} + 2\hat{j} - 3\hat{k} \]

### Step 2: Write the vector equation of the line
The vector equation of a line in space can be written in the form:
\[ \mathbf{R} = \mathbf{A} + \lambda \mathbf{B} \]
where \( \mathbf{R} \) is the position vector of any point on the line, \( \mathbf{A} \) is the position vector of a point on the line, \( \lambda \) is a scalar parameter, and \( \mathbf{B} \) is the direction vector.

Substituting the values of \( \mathbf{A} \) and \( \mathbf{B} \):
\[ \mathbf{R} = (3\hat{i} + 4\hat{j} + 5\hat{k}) + \lambda (2\hat{i} + 2\hat{j} - 3\hat{k}) \]

### Step 3: Simplify the equation
Distributing \( \lambda \) in the equation:
\[ \mathbf{R} = 3\hat{i} + 4\hat{j} + 5\hat{k} + \lambda (2\hat{i} + 2\hat{j} - 3\hat{k}) \]
This can be rewritten as:
\[ \mathbf{R} = (3 + 2\lambda)\hat{i} + (4 + 2\lambda)\hat{j} + (5 - 3\lambda)\hat{k} \]

### Final Vector Equation
Thus, the vector equation of the line is:
\[ \mathbf{R} = (3 + 2\lambda)\hat{i} + (4 + 2\lambda)\hat{j} + (5 - 3\lambda)\hat{k} \]