Question

The vector equation of the line passing through the point having position vector \( 4\hat{i} - \hat{j} + 2\hat{k} \) and parallel to vector \( -2\hat{i} - \hat{j} + \hat{k} \) is given by .........

• A. \( (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda(-2\hat{i} - \hat{j} + \hat{k}) \)
• B. \( (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda(2\hat{i} - \hat{j} + \hat{k}) \)
• C. \( (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda(-2\hat{i} - \hat{j} - \hat{k}) \)
• D. \( (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda(-2\hat{i} - \hat{j} + \hat{k}) \)

Hint:

Remember, the vector equation of a line involves a point and a direction vector.

Answer 1

The Correct Option is D

Step 1: Recall the vector equation of a line.
The vector equation of a line passing through a point \(\vec{r_0}\) and parallel to a direction vector \(\vec{d}\) is given by: \[ \vec{r} = \vec{r_0} + \lambda \vec{d} \]

Step 2: Apply the values.
Here, the position vector \(\vec{r_0} = 4\hat{i} - \hat{j} + 2\hat{k}\), and the direction vector \(\vec{d} = -2\hat{i} - \hat{j} + \hat{k}\).

Step 3: Write the equation.
The vector equation is: \[ \vec{r} = (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda(-2\hat{i} - \hat{j} + \hat{k}) \]

Step 4: Conclude.
Thus, the correct equation is option (iv).

Final Answer: \[ \boxed{(4\hat{i} - \hat{j} + 2\hat{k}) + \lambda(-2\hat{i} - \hat{j} + \hat{k})} \]