1. Key Concepts and Formulas

• Skew Lines: In three-dimensional geometry, two lines are considered skew if they are neither parallel nor intersecting. They lie in different planes.
• Vector Equation of a Line: A line passing through a point with position vector $\vec{a}$ and parallel to a direction vector $\vec{b}$ can be represented as $\vec{r} = \vec{a} + \lambda \vec{b}$, where $\lambda$ is a scalar parameter.
• Shortest Distance Formula: The shortest distance $d$ between two skew lines $L_1: \vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $L_2: \vec{r} = \vec{a}_2 + \mu \vec{b}_2$ is given by the formula: $d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$ This formula calculates the projection of the vector connecting a point on $L_1$ to a point on $L_2$ onto the direction vector that is perpendicular to both lines (i.e., $\vec{b}_1 \times \vec{b}_2$).

2. Step-by-Step Solution

Step 1: Identify Position and Direction Vectors for Each Line The given lines are in Cartesian form. We first extract the position vector ($\vec{a}$) of a point on the line and the direction vector ($\vec{b}$) for each line.

For $L_1: \frac{x-3}{2}=\frac{y+15}{-7}=\frac{z+3}{5}$

• The line passes through the point $(3, -15, -3)$. Therefore, the position vector $\vec{a}_1 = 3\hat{i} - 15\hat{j} - 3\hat{k}$.
• The direction ratios are $(2, -7, 5)$. Therefore, the direction vector $\vec{b}_1 = 2\hat{i} - 7\hat{j} + 5\hat{k}$.

For $L_2: \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$

• The line passes through the point $(-1, 1, 9)$. Therefore, the position vector $\vec{a}_2 = -1\hat{i} + 1\hat{j} + 9\hat{k}$.
• The direction ratios are $(2, 1, -3)$. Therefore, the direction vector $\vec{b}_2 = 2\hat{i} + 1\hat{j} - 3\hat{k}$.

Step 2: Check for Parallelism (Optional but Recommended) It's good practice to check if the lines are parallel. If $\vec{b}_1$ is a scalar multiple of $\vec{b}_2$ (i.e., $\vec{b}_1 = k\vec{b}_2$ for some scalar $k$), the lines are parallel. Here, $\vec{b}_1 = (2, -7, 5)$ and $\vec{b}_2 = (2, 1, -3)$. Comparing the ratios of corresponding components: $\frac{2}{2} = 1$, but $\frac{-7}{1} = -7$. Since these ratios are not equal, the direction vectors are not parallel, which means the lines are not parallel. Thus, they are either skew or intersecting, and the shortest distance formula for skew lines is appropriate.

Step 3: Calculate the Vector Connecting Points on the Lines: $\vec{a}_2 - \vec{a}_1$ This vector represents the displacement from a point on $L_1$ to a point on $L_2$. $\vec{a}_2 - \vec{a}_1 = (-1\hat{i} + 1\hat{j} + 9\hat{k}) - (3\hat{i} - 15\hat{j} - 3\hat{k})$ $\vec{a}_2 - \vec{a}_1 = (-1-3)\hat{i} + (1-(-15))\hat{j} + (9-(-3))\hat{k}$ $\vec{a}_2 - \vec{a}_1 = -4\hat{i} + 16\hat{j} + 12\hat{k}$

Step 4: Calculate the Cross Product of Direction Vectors: $\vec{b}_1 \times \vec{b}_2$ This cross product yields a vector that is perpendicular to both $\vec{b}_1$ and $\vec{b}_2$. This direction is crucial as it represents the unique line segment of shortest distance. $\vec{b}_1 \times \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -7 & 5 \\ 2 & 1 & -3 \end{vmatrix}$ $= \hat{i}((-7)(-3) - (5)(1)) - \hat{j}((2)(-3) - (5)(2)) + \hat{k}((2)(1) - (-7)(2))$ $= \hat{i}(21 - 5) - \hat{j}(-6 - 10) + \hat{k}(2 + 14)$ $= 16\hat{i} - (-16)\hat{j} + 16\hat{k}$ $= 16\hat{i} + 16\hat{j} + 16\hat{k}$

Step 5: Calculate the Magnitude of the Cross Product: $|\vec{b}_1 \times \vec{b}_2|$ This magnitude forms the denominator of our shortest distance formula. $|\vec{b}_1 \times \vec{b}_2| = |16\hat{i} + 16\hat{j} + 16\hat{k}|$ $= \sqrt{(16)^2 + (16)^2 + (16)^2}$ $= \sqrt{3 \times (16)^2}$ $= 16\sqrt{3}$

Step 6: Calculate the Scalar Triple Product: $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)$ This dot product forms the numerator of our formula (its absolute value). $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = (-4\hat{i} + 16\hat{j} + 12\hat{k}) \cdot (16\hat{i} + 16\hat{j} + 16\hat{k})$ $= (-4)(16) + (16)(16) + (12)(16)$ $= -64 + 256 + 192$ $= 192 + 192$ $= 384$ We need the absolute value of this result for the distance, which is $|384| = 384$.

Step 7: Calculate the Shortest Distance Now, substitute the calculated values into the shortest distance formula: $d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$ $d = \frac{384}{16\sqrt{3}}$ To simplify, first divide 384 by 16: $d = \frac{24}{\sqrt{3}}$ Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$: $d = \frac{24\sqrt{3}}{\sqrt{3} \times \sqrt{3}}$ $d = \frac{24\sqrt{3}}{3}$ $d = 8\sqrt{3}$

3. Common Mistakes & Tips

• Sign Errors in Point Extraction: Be extremely careful when extracting the coordinates of the points from the Cartesian form. For instance, $y+15$ implies $y-(-15)$, so the y-coordinate is $-15$. Similarly, $z+3$ implies $z-(-3)$, so the z-coordinate is $-3$.
• Cross Product Calculation: Errors in calculating the determinant for the cross product are common. Double-check your arithmetic, especially the signs.
• Absolute Value: Always remember that distance is a non-negative quantity, so take the absolute value of the scalar triple product in the numerator.
• Parallelism Check: Although not strictly necessary for skew lines, a quick check for parallelism can prevent using the wrong formula if the lines happen to be parallel.

4. Summary The problem involves finding the shortest distance between two skew lines. This is achieved by first identifying the position vectors of points on each line and their respective direction vectors. Then, we calculate the vector connecting these points, the cross product of the direction vectors, and their magnitudes. Finally, the scalar triple product is computed and divided by the magnitude of the cross product, yielding the shortest distance. Following these precise vector operations, the shortest distance between the given lines is found to be $8\sqrt{3}$ units.

5. Final Answer The final answer is \boxed{\text{8 \sqrt{3}}}, which corresponds to option (A).