Question

Let the line of the shortest distance between the lines L 1 : r ⃗ = ( i ^ + 2 j ^ + 3 k ^ ) + λ ( i ^ − j ^ + k ^ ) and L 2 : r ⃗ = ( 4 i ^ + 5 j ^ + 6 k ^ ) + μ ( i ^ + j ^ − k ^ ) intersect L 1 and L 2 at P and Q, respectively. If ( α , β , γ ) is the midpoint of the line segment P Q, then 2 ( α + β + γ ) is equal to

• A. 21
• B. 10.5
• C. 42
• D. 84

Answer 1

Answer: (A)

21

Answer 2

1. Calculate the direction vector $\vec{d}$ of the shortest distance line using the cross product of the direction vectors of the given lines.
2. Represent points P and Q on L1 and L2 using parameters $\lambda$ and $\mu$.
3. Form the vector $\vec{PQ}$ and use the conditions that $\vec{PQ}$ is perpendicular to $\vec{b_1}$ and $\vec{b_2}$ to set up a system of equations for $\lambda$ and $\mu$.
4. Solve for $\lambda$ and $\mu$ to find the coordinates of P and Q.
5. Calculate the midpoint $(\alpha, \beta, \gamma)$ of PQ.
6. Compute the required value $2(\alpha + \beta + \gamma)$.