Find the vector equations of the line passing through the point having positions vector ` -hati -hatj +2hatk ` and parallel to the line ` overset(-) r = ( hati+ 2hatj + 3hatk ) +mu (3hati + 2hatj +hatk) , mu ` is a parameter .

Answer

To find the vector equation of the line passing through the point with position vector \(-\hat{i} - \hat{j} + 2\hat{k}\) and parallel to the line given by \(\overset{-}{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu (3\hat{i} + 2\hat{j} + \hat{k})\), we can follow these steps:

### Step 1: Identify the point and direction vector
The point through which the line passes is given by the position vector:
\[ \mathbf{A} = -\hat{i} - \hat{j} + 2\hat{k} \]
The direction vector of the given line can be extracted from the equation:
\[ \mathbf{B} = 3\hat{i} + 2\hat{j} + \hat{k} \]

### Step 2: Write the vector equation of the line
The vector equation of a line can be expressed in the form:
\[ \mathbf{r} = \mathbf{A} + \lambda \mathbf{B} \]
where \(\lambda\) is a parameter.

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

### Step 3: Simplify the equation
Now we can simplify this equation:
\[ \mathbf{r} = -\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 2\hat{j} + \hat{k}) \]
This can be rewritten as:
\[ \mathbf{r} = (-1 + 3\lambda)\hat{i} + (-1 + 2\lambda)\hat{j} + (2 + \lambda)\hat{k} \]

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

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