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

Updated On: Mar 12, 2026

$5\sqrt 3$
$2\sqrt 3$
$3\sqrt 3$
$4\sqrt 3$

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The Correct Option is D

Solution and Explanation

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.

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Concepts Used:

Coordinate Geometry

Coordinate geometry, also known as analytical geometry or Cartesian geometry, is a branch of mathematics that combines algebraic techniques with the principles of geometry. It provides a way to represent geometric figures and solve problems using algebraic equations and coordinate systems.
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.