Find the vector equation of a line passes through the point `hati+3hatj+hatk` and parallel to vector `3hati-2hatj+k`.

Answer

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

### Step 1: Identify the point and direction vector
The point through which the line passes is given as:
\[ \mathbf{A} = \hat{i} + 3\hat{j} + \hat{k} \]
The direction vector, which is parallel to the line, is given as:
\[ \mathbf{B} = 3\hat{i} - 2\hat{j} + \hat{k} \]

### Step 2: Write the vector equation of the line
The vector equation of a line that passes through a point \( \mathbf{A} \) and is parallel to a vector \( \mathbf{B} \) can be expressed as:
\[ \mathbf{R} = \mathbf{A} + \lambda \mathbf{B} \]
where \( \lambda \) is a scalar parameter.

### Step 3: Substitute the values of \( \mathbf{A} \) and \( \mathbf{B} \)
Now, substituting the values of \( \mathbf{A} \) and \( \mathbf{B} \) into the equation:
\[ \mathbf{R} = (\hat{i} + 3\hat{j} + \hat{k}) + \lambda (3\hat{i} - 2\hat{j} + \hat{k}) \]

### Step 4: Expand the equation
Expanding the equation gives:
\[ \mathbf{R} = \hat{i} + 3\hat{j} + \hat{k} + \lambda(3\hat{i}) - \lambda(2\hat{j}) + \lambda(\hat{k}) \]
\[ \mathbf{R} = \hat{i} + 3\hat{j} + \hat{k} + 3\lambda\hat{i} - 2\lambda\hat{j} + \lambda\hat{k} \]

### Step 5: Combine like terms
Combining the like terms results in:
\[ \mathbf{R} = (1 + 3\lambda)\hat{i} + (3 - 2\lambda)\hat{j} + (1 + \lambda)\hat{k} \]

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