Find the vector equations of the line passing through the point having positions vector ` 4 hati -hatj +2hatk ` and parallel to the vector` -2hati -hatj +hatk `

Answer

To find the vector equation of the line passing through a given point and parallel to a specified vector, we can follow these steps:

### Step 1: Identify the position vector of the point
The position vector of the point through which the line passes is given as:
\[ \mathbf{a} = 4\hat{i} - \hat{j} + 2\hat{k} \]

### Step 2: Identify the direction vector
The vector that is parallel to the line is given as:
\[ \mathbf{b} = -2\hat{i} - \hat{j} + \hat{k} \]

### Step 3: 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 \(\mathbf{r}\) is the position vector of any point on the line, \(\lambda\) is a scalar parameter, \(\mathbf{a}\) is the position vector of a point on the line, and \(\mathbf{b}\) is the direction vector.

Substituting the values of \(\mathbf{a}\) and \(\mathbf{b}\) into the equation, we get:
\[ \mathbf{r} = (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda (-2\hat{i} - \hat{j} + \hat{k}) \]

### Step 4: Simplify the equation
Distributing \(\lambda\) and combining like terms:
\[ \mathbf{r} = 4\hat{i} - \hat{j} + 2\hat{k} - 2\lambda\hat{i} - \lambda\hat{j} + \lambda\hat{k} \]
\[ \mathbf{r} = (4 - 2\lambda)\hat{i} + (-1 - \lambda)\hat{j} + (2 + \lambda)\hat{k} \]

### Final Answer
Thus, the vector equation of the line is:
\[ \mathbf{r} = (4 - 2\lambda)\hat{i} + (-1 - \lambda)\hat{j} + (2 + \lambda)\hat{k} \]

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