1. Key Concepts and Formulas

• Parametric Form of a Line: A line passing through $(x_0, y_0, z_0)$ with direction vector $(a,b,c)$ can be represented as $\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c} = \mu$, yielding a general point $(x_0+a\mu, y_0+b\mu, z_0+c\mu)$. For a vector form $\vec{r} = \vec{a} + \lambda \vec{d}$, a general point is $(\vec{a} + \lambda \vec{d})$.
• Foot of the Perpendicular: If P is the foot of the perpendicular from point A to line L, then the vector $\vec{AP}$ is perpendicular to the direction vector of line L, $\vec{d_L}$. This implies their dot product is zero: $\vec{AP} \cdot \vec{d_L} = 0$.
• Intersection of Two Lines: Two lines intersect if there exists a common point that satisfies the parametric equations of both lines. This involves setting the coordinates of the general points on each line (using different parameters) equal to each other and solving the resulting system of equations.
• Distance Formula in 3D: The square of the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $D^2 = (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2$.

2. Step-by-Step Solution

Part 1: Finding P, the foot of the perpendicular from point A to line L.

Let the given point be $A(1,2,2)$. The line L is given by $\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$.

• Step 1: Parametrize a general point on Line L. We introduce a parameter, say $\mu$, to represent any point on line L. $\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}=\mu$ This gives the coordinates of a general point $P_{gen}$ on L as: $P_{gen}(\mu+1, -\mu-1, 2\mu+2)$

• Step 2: Form the vector $\vec{AP}$. The vector $\vec{AP}$ connects the external point $A(1,2,2)$ to the general point $P_{gen}$ on line L. $\vec{AP} = ((\mu+1)-1, (-\mu-1)-2, (2\mu+2)-2)$ $\vec{AP} = (\mu, -\mu-3, 2\mu)$

• Step 3: Apply the perpendicularity condition. The direction vector of line L, $\vec{d_L}$, is obtained from the denominators of its Cartesian equation: $\vec{d_L} = (1, -1, 2)$. Since P is the foot of the perpendicular, the vector $\vec{AP}$ must be perpendicular to $\vec{d_L}$. Their dot product must be zero. $\vec{AP} \cdot \vec{d_L} = 0$ $(\mu)(1) + (-\mu-3)(-1) + (2\mu)(2) = 0$ $\mu + (\mu+3) + 4\mu = 0$ $6\mu + 3 = 0$ $\mu = -\frac{1}{2}$

• Step 4: Calculate the coordinates of P. Substitute the value $\mu = -\frac{1}{2}$ back into the general point $P_{gen}(\mu+1, -\mu-1, 2\mu+2)$:

  • $x_P = (-\frac{1}{2})+1 = \frac{1}{2}$
  • $y_P = -(-\frac{1}{2})-1 = \frac{1}{2}-1 = -\frac{1}{2}$
  • $z_P = 2(-\frac{1}{2})+2 = -1+2 = 1$ Thus, the foot of the perpendicular is $P\left(\frac{1}{2}, -\frac{1}{2}, 1\right)$.

Part 2: Finding Q, the intersection point of line L and line L'.

Line L is $P_{gen}(\mu+1, -\mu-1, 2\mu+2)$. Line L' is given by $\vec{r}=(-\hat{i}+\hat{j}-2 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$.

• Step 1: Parametrize a general point on Line L'. The vector form directly gives the parametric equations for any point on L' in terms of $\lambda$: $Q_{gen}(-1+\lambda, 1-\lambda, -2+\lambda)$

• Step 2: Equate coordinates for intersection. For the lines to intersect at point Q, the coordinates of $P_{gen}$ and $Q_{gen}$ must be identical for specific values of $\mu$ and $\lambda$.

  1. x-coordinates: $\mu+1 = -1+\lambda \quad \Rightarrow \quad \mu - \lambda = -2 \quad \text{(Eq. 1)}$
  2. y-coordinates: $-\mu-1 = 1-\lambda \quad \Rightarrow \quad -\mu + \lambda = 2 \quad \text{(Eq. 2)}$
  3. z-coordinates: $2\mu+2 = -2+\lambda \quad \Rightarrow \quad 2\mu - \lambda = -4 \quad \text{(Eq. 3)}$

• Step 3: Solve the system of equations for $\mu$ and $\lambda$. Notice that (Eq. 2) is simply $(-1) \times \text{(Eq. 1)}$, indicating dependency. We use (Eq. 1) and (Eq. 3) to find the unique values of $\mu$ and $\lambda$. Subtract (Eq. 1) from (Eq. 3): $(2\mu - \lambda) - (\mu - \lambda) = -4 - (-2)$ $\mu = -2$ Substitute $\mu = -2$ into (Eq. 1): $(-2) - \lambda = -2$ $-\lambda = 0 \quad \Rightarrow \quad \lambda = 0$

• Step 4: Verify consistency. It's crucial to check if these values satisfy all three original equations. We verify with (Eq. 2): $-(-\mu) + \lambda = 2$ $-(-2) + 0 = 2$ $2 = 2$ The values $\mu = -2$ and $\lambda = 0$ are consistent, so the lines intersect.

• Step 5: Calculate the coordinates of Q. Substitute $\lambda = 0$ into the general point $Q_{gen}(-1+\lambda, 1-\lambda, -2+\lambda)$:

  • $x_Q = -1+0 = -1$
  • $y_Q = 1-0 = 1$
  • $z_Q = -2+0 = -2$ The intersection point is $Q(-1, 1, -2)$.

Part 3: Calculating $2(PQ)^2$.

We have the coordinates of P and Q: $P\left(\frac{1}{2}, -\frac{1}{2}, 1\right)$ $Q(-1, 1, -2)$

• Step 1: Calculate the square of the distance $(PQ)^2$. Using the 3D distance formula $D^2 = (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2$: $(PQ)^2 = \left(-1 - \frac{1}{2}\right)^2 + \left(1 - \left(-\frac{1}{2}\right)\right)^2 + (-2 - 1)^2$ $(PQ)^2 = \left(-\frac{3}{2}\right)^2 + \left(\frac{3}{2}\right)^2 + (-3)^2$ $(PQ)^2 = \frac{9}{4} + \frac{9}{4} + 9$ $(PQ)^2 = \frac{18}{4} + 9$ $(PQ)^2 = \frac{9}{2} + \frac{18}{2}$ $(PQ)^2 = \frac{27}{2}$

• Step 2: Calculate $2(PQ)^2$. The problem asks for $2(PQ)^2$: $2(PQ)^2 = 2 \times \frac{27}{2}$ $2(PQ)^2 = 27$

3. Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when substituting coordinates or performing subtractions, especially with negative numbers.
• Parameter Confusion: Always use different parameters (e.g., $\mu$ and $\lambda$) for different lines when finding their intersection to avoid errors.
• Consistency Check: After solving for the parameters of intersection, always verify them in all three coordinate equations. If they don't satisfy all equations, the lines are skew and do not intersect.
• Substitution: Remember to substitute the calculated parameter values back into the general point's coordinates to find the specific points P and Q.

4. Summary

This problem was a comprehensive application of 3D geometry concepts. We systematically determined the coordinates of point P, the foot of the perpendicular from a given point to line L, by using the dot product condition for perpendicularity. Subsequently, we found point Q, the intersection of line L and another line L', by equating their parametric coordinate expressions and solving the resulting system of linear equations. Finally, with the coordinates of P and Q established, we applied the 3D distance formula to calculate $(PQ)^2$ and then multiplied by 2 to arrive at the final required value.

5. Final Answer

The final answer is $\boxed{27}$, which corresponds to option (A).