Question

The shortest distance between the lines $\frac{x−1}2 =\frac{y−2}1 =\frac{z−6}{-3} $ and $\frac{x−1}2 =\frac{y+8}{-7} =\frac{z−4}{5} $ is

• A. \(5\sqrt 3\)
• B. \(2\sqrt 3\)
• C. \(3\sqrt 3\)
• D. \(4\sqrt 3\)

Answer 1

The Correct Option is D

Step 1: Find the direction vectors from the parametric equations.

The direction vector for the first equation is a = î - 8ĵ + 4k̂.

The direction vector for the second equation is b = î + 2ĵ + 6k̂.

Step 2: Find the cross product of the two vectors p × q.

p × q = 2î - 7ĵ + 5k̂  (from first vector)
             2î + ĵ - 3k̂  (from second vector)

    The cross product is:
    p × q = î(16) - ĵ(16) + k̂(16)
          = 16(î + ĵ + k̂)
    

Step 3: Find the magnitude of a - b divided by the magnitude of p × q.

d = |a - b| * |p × q| / |p × q|
        = |-10ĵ - 2k̂| * |16(î + ĵ + k̂)| / (16√3)
        = |-12/√3| = 4√3
    

Final Answer: The value of d is 4√3.