Question

Find the vector equation of the line passing through the point having position vector \[ 4\hat{i} - \hat{j} + 2\hat{k} \] and parallel to the vector \[ -2\hat{i} - \hat{j} + \hat{k}. \]

Hint:

The vector equation of a line through \( \mathbf{r_0} \) parallel to \( \mathbf{d} \) is: \[ \mathbf{r} = \mathbf{r_0} + \lambda \mathbf{d}. \]

Answer 1

Step 1: Vector Equation of a Line
The vector equation of a line passing through \( \mathbf{r_0} \) and parallel to \( \mathbf{d} \) is: \[ \mathbf{r} = \mathbf{r_0} + \lambda \mathbf{d}. \] Step 2: Substitute Given Vectors
\[ \mathbf{r_0} = 4\hat{i} - \hat{j} + 2\hat{k}, \quad \mathbf{d} = -2\hat{i} - \hat{j} + \hat{k}. \] Step 3: Write the Equation
\[ \mathbf{r} = (4\hat{i} - \hat{j} + 2\hat{k}) + \lambda (-2\hat{i} - \hat{j} + \hat{k}). \] Expanding: \[ \mathbf{r} = (4 - 2\lambda) \hat{i} + (-1 - \lambda) \hat{j} + (2 + \lambda) \hat{k}. \]