Check whether the lines given by $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 4}{5} = \frac{y - 1}{2} = z$ are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

Three Dimensional Geometry

Lines in Three-Dimensional Geometry — Parallelism and Intersection

Official Answer

Compare direction ratios of both lines to test parallelism, then solve simultaneously for the point of intersection if not parallel.

Testing parallelism

Line 1 passes through $(1,2,3)$ with direction ratios $(2,3,4)$ ; Line 2 passes through $(4,1,0)$ with direction ratios $(5,2,1)$ .
Since $2/5 \ne 3/2 \ne 4/1$ , the direction ratios are not proportional, so the lines are not parallel.

Finding the point of intersection

Writing both lines in parametric form and equating coordinates gives three equations in parameters s and t, which are found to be mutually consistent ( $s = -1, t = -1$ ).
Substituting back gives the common point $(-1, -1, -1)$ , confirming the two lines intersect at this point.

direction ratiosparallel lines in spaceintersecting linesskew linesequation of a line in 3Dpoint of intersectionparametric form

Marking Scheme

11 mark: correctly identifying points $(1,2,3)$ , $(4,1,0)$ and direction ratios $(2,3,4)$ , $(5,2,1)$ of the two lines.
21 mark: correctly checking $2/5 \ne 3/2 \ne 4/1$ and concluding the lines are not parallel.
31 mark: writing both lines in parametric form using parameters s and t.
41 mark: solving any two equations to get $s=-1, t=-1$ .
51 mark: verifying the third equation and stating the point of intersection $(-1,-1,-1)$ .

Hint

Compare direction ratios $(2,3,4)$ and $(5,2,1)$ for proportionality; if not parallel, equate parametric coordinates of both lines and solve for the two parameters, then verify consistency in the third equation.

Quick Oral Answer

The direction ratios $(2,3,4)$ and $(5,2,1)$ are not proportional, so the lines aren't parallel; solving the parametric equations shows they meet consistently at the point $(-1,-1,-1)$ .

Analysis & Explanation

This question tests the ability to distinguish between parallel, intersecting, and skew lines in 3D — a core theme of the Three Dimensional Geometry chapter.

Concept

Two lines are parallel if and only if their direction ratios are proportional ( $a_1/a_2 = b_1/b_2 = c_1/c_2$ ).
If not parallel, lines in space are either intersecting or skew; to check, write both lines parametrically and solve any two of the three coordinate equations for the parameters, then verify the third equation is also satisfied — if it is, the lines intersect (skew lines would give an inconsistency in the third equation).

Exam trap

A common mistake is stopping after checking parallelism without proceeding to check intersection when the lines are not parallel — the question explicitly requires this next step.
Another frequent error is solving only two of the three parametric equations and assuming intersection without verifying the third equation, which risks concluding "intersecting" for what are actually skew lines.

Real-world relevance

Determining whether two lines (e.g., flight paths, cable routes, or laser beams) intersect, are parallel, or are skew is fundamental in aerospace trajectory planning, robotics path-collision checks, and computer graphics ray-tracing algorithms.

Common Mistakes

1Concluding the lines are parallel merely because both are given in symmetric form, without actually checking proportionality of direction ratios.
2Solving only two of the three coordinate equations and declaring intersection without checking the third equation for consistency (risk of wrongly calling skew lines 'intersecting').
3Arithmetic sign errors while equating parametric coordinates, leading to an incorrect point of intersection.

Interesting Facts

Skew lines — lines that are neither parallel nor intersecting — can only exist in three or more dimensions; in 2D, any two distinct non-parallel lines must always intersect.

The formula for shortest distance between skew lines (used when lines are neither parallel nor intersecting) was central to 19th-century work in projective and analytic geometry by mathematicians like Julius Plücker.

Ray-tracing algorithms in 3D computer graphics rely on exactly this line-intersection test billions of times per rendered frame to determine which surfaces a light ray hits.

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Frequently Asked Questions

How do you check if two lines in 3D are parallel?

Compare their direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$ ; the lines are parallel if $a_1/a_2 = b_1/b_2 = c_1/c_2$ .

What if the two lines are neither parallel nor intersecting?

Then they are skew lines, and instead of a point of intersection, we compute the shortest distance between them using the skew-line distance formula.

Why must all three parametric equations be checked?

Solving only two of the three equations only guarantees a solution in 2 coordinates; the third equation must also be satisfied for the lines to actually meet at a common 3D point, otherwise they are skew.