Step 1: Understanding the Question:
Find the intersection points of a line with coordinate planes and then find the equation of the line connecting those two points.
Step 2: Key Formula or Approach:
1. XY plane is $z=0$. YZ plane is $x=0$.
2. Vector equation $(\vec{r} - \vec{a}) \times \vec{b} = \vec{0}$ represents a line passing through $\vec{a}$ in the direction of $\vec{b}$.
Step 3: Detailed Explanation:
General point on the line is $(2\lambda + 1, -\lambda - 2, \lambda)$.
For point A (intersects XY plane $\implies z = 0$):
$\lambda = 0 \implies A = (2(0)+1, -0-2, 0) = (1, -2, 0)$.
For point B (intersects YZ plane $\implies x = 0$):
$2\lambda + 1 = 0 \implies \lambda = -1/2$.
$B = (0, -(-1/2)-2, -1/2) = (0, -3/2, -1/2)$.
Direction vector of line AB:
$\vec{AB} = \vec{b} - \vec{a} = (0-1, -3/2 - (-2), -1/2 - 0) = (-1, 1/2, -1/2)$.
The line passes through $A(1, -2, 0)$, so $\vec{a} = \hat{i} - 2\hat{j} + 0\hat{k}$.
Using the form $(\vec{r} - \vec{a}) \times \vec{d} = \vec{0}$:
$[\vec{r} - (\hat{i} - 2\hat{j} + 0\hat{k})] \times (-\hat{i} + \frac{1}{2}\hat{j} - \frac{1}{2}\hat{k}) = \vec{0}$.
Step 4: Final Answer:
The equation is option (A).