Mathematics

Mathematics

Problem

Match the following
Column IColumn II
A. x−12=y−23=z−34\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} and x−13=y−34=z−55\frac{x-1}{3} = \frac{y-3}{4} = \frac{z-5}{5} arep. coincident
B. x−12=y−23=z−34\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} and x−32=y−53=z−74\frac{x-3}{2} = \frac{y-5}{3} = \frac{z-7}{4} areq. parallel and different
C. x−25=y+34=z−22\frac{x-2}{5} = \frac{y+3}{4} = \frac{z-2}{2} and x−75=y−14=z−2−2\frac{x-7}{5} = \frac{y-1}{4} = \frac{z-2}{-2} arer. skew
D. x−32=y+23=z−45\frac{x-3}{2} = \frac{y+2}{3} = \frac{z-4}{5} and x−33=y−22=z−75\frac{x-3}{3} = \frac{y-2}{2} = \frac{z-7}{5} ares. intersecting in a point
Intuitive explanation

Lines in 3D space can be classified by comparing their direction vectors and checking whether they share a common point. Extract the direction vector and a point on each line from the symmetric form, then test: if direction vectors are proportional, the lines are parallel (check if they're the same line or distinct); if not parallel, check if they intersect by solving the parametric equations, or if they're skew (non-parallel and non-intersecting). This classification exhausts all possibilities for lines in space.

Intuitive explanation

Lines in 3D space can be classified by comparing their direction vectors and checking whether they share a common point. Extract the direction vector and a point on each line from the symmetric form, then test: if direction vectors are proportional, the lines are parallel (check if they're the same line or distinct); if not parallel, check if they intersect by solving the parametric equations, or if they're skew (non-parallel and non-intersecting). This classification exhausts all possibilities for lines in space.

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