Question

If the Cartesian equation of the line is \[ x - 1 = 2y + 3 = 3 - z, \] then its vector equation is

• A. \( \vec{r} = \hat{i} - 3\hat{j} + 3\hat{k} + \lambda (2\hat{i} + \hat{j} - 2\hat{k}) \)
• B. \( \vec{r} = -\hat{i} - 3\hat{j} + 3\hat{k} + \lambda (\hat{i} + \frac{1}{2} \hat{j} - \hat{k}) \)
• C. \( \vec{r} = -\hat{i} - \frac{3}{2} \hat{j} - 3\hat{k} + \lambda (2\hat{i} + \hat{j} - 2\hat{k}) \)
• D. \( \vec{r} = \hat{i} - \frac{3}{2} \hat{j} + 3\hat{k} + \lambda (2\hat{i} + \hat{j} - 2\hat{k}) \)

Hint:

To convert a Cartesian equation to a vector equation, first convert it to parametric form, then express it as a vector equation.

Answer 1

The Correct Option is D

Step 1: Convert the Cartesian equation to parametric form.
The given Cartesian equation can be written as three equations: \[ x - 1 = 2y + 3 = 3 - z = \lambda. \] From this, we obtain: \[ x = \lambda + 1, \quad y = \frac{\lambda - 3}{2}, \quad z = 3 - \lambda. \]
Step 2: Write the vector equation.
The parametric form of the equation can be written as: \[ \vec{r} = \hat{i} - \frac{3}{2} \hat{j} + 3 \hat{k} + \lambda (2 \hat{i} + \hat{j} - 2 \hat{k}). \]
Step 3: Conclusion.
Thus, the vector equation of the line is \( \vec{r} = \hat{i} - \frac{3}{2} \hat{j} + 3 \hat{k} + \lambda (2 \hat{i} + \hat{j} - 2 \hat{k}) \), which corresponds to option (D).