Question

The vector equation of a line passing through the point \( (1, -1, 0) \) and parallel to the \( y \)-axis is:

• A. \[ \mathbf{r} = \hat{i} - \hat{j} + \lambda (\hat{i} - \hat{j}) \]
• B. \[ \mathbf{r} = \hat{i} - \hat{j} + \lambda \hat{j} \]
• C. \[ \mathbf{r} = \hat{i} - \hat{j} + \lambda \hat{k} \]
• D. \[ \mathbf{r} = \lambda \hat{j} \]

Hint:

Vector equations of lines require a point and a direction vector.

Answer 1

The Correct Option is B

Step 1: Determine the direction vector.
The direction vector lies along the \( y \)-axis, represented as \( \hat{j} \).

Step 2: Formulate the line equation.
The vector equation for a line passing through point \( (1, -1, 0) \) is given by: \[ \mathbf{r} = \mathbf{r}_0 + \lambda \mathbf{d}, \] where \( \mathbf{r}_0 \) is the position vector of the point \( \hat{i} - \hat{j} \) and \( \mathbf{d} \) is the direction vector \( \hat{j} \).

Step 3: Substitute the determined values.
\[ \mathbf{r} = (\hat{i} - \hat{j}) + \lambda \hat{j} = \hat{i} - \hat{j} + \lambda \hat{j}. \]
Step 4: Validate against provided options.
The derived equation is: \[ \mathbf{r} = \hat{i} - \hat{j} + \lambda \hat{j} \] This equation corresponds to option (B).