Math notesVectors & 3D3D Lines
Independent topic page
3D Lines full marks = write direction vector b first, then choose angle / intersect / skew-SD / coplanar gate — never use 2D distance intuition on skew lines.
01. Line forms: symmetric (x−x₀)/l=(y−y₀)/m=(z−z₀)/n and vector r=a+λb. Extract b (DRs) before any formula.
02. Two lines: parallel (b₁∥b₂), intersecting/coplanar, or skew. Gate with (a₂−a₁)·(b₁×b₂) before calling skew.
03. Skew SD = |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|. If numerator is 0 and b₁×b₂≠0, lines intersect (coplanar non-parallel).
04. Full marks: a₁,b₁,a₂,b₂ on rough sheet in four labels — then one formula.
05. Advanced lane: every line tool starts from direction ratios/cosines of b — write DR before distance, angle, or skew formulae. DC vs raw DR: l²+m²+n²=1 only for cosines.
06. Skew vs intersecting gate: check coplanarity / shortest-distance numerator before applying the skew SD formula to lines that actually meet.
07. Two-line formulae: label a₁,b₁ and a₂,b₂ on the rough sheet first — most numerical errors are swapped plugs, not wrong identities.
Official · 2026
Official syllabus points for Lines in Three Dimensions only — not the full Vectors & 3D portion map. Portion hub: Vectors & 3D · Vectors & 3D analysis.
No topic-specific exclusions flagged — finish the unit points below for 3D Lines.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusDirection cosines/ratios, skew lines, and shortest distance — not the full planes unit.
Syllabus points
2020–2026 paper weight
Topic match inside parent portion Vectors & 3D (11.2% of Math tags). Stats below are portion backdrop; matched subtopics show where Lines in Three Dimensions sits.
Portion-wide type/difficulty mix lives on the Vectors & 3D analysis — not repeated here so this page stays on 3D Lines.
01. Dominant subtopics: 3D Geometry (15), Vectors (12).
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Lines in Three Dimensions — practise the knowledge bridge, not only the single-chapter drill.
Step 1: solve x y' = y − x³ (homogeneous/linear in y/x) to get the family y = Cx − x³/2. Step 2: impose the given condition to fix C, then use y' = 0 to locate the local max. Step 3: compare monotonicity on (1,2) and count positive roots of f = g.
Multi-correct options test DE, calculus and graph reading together; one sign error in the DE kills several options.
After every DE, immediately ask: extrema? monotonicity? intersections with a given g?
Step 1: extract all shared geometric data from the paragraph once. Step 2: answer the first NUM (slope/intercept/distance). Step 3: reuse intermediates for the linked follow-ups without re-deriving from scratch.
Early example of Advanced’s linked-stem tax on organisation.
Box shared quantities at the top of linked stems — treat them as a mini formula sheet.
Why methods matter
3D lines are b first, then gate skew vs intersecting. These methods stop 2D intuition on skew SD.
Every line formula is built from b — write it before the sheet of identities.
Do thisAlways write b (direction) before angle or SD formulae.
Every line formula is built from b.
Angle between lines = angle between direction vectors.
Prevents the classic wrong use of skew SD on intersecting lines.
Do thisCompute (a₂−a₁)·(b₁×b₂) before calling lines skew.
Stops wrong use of SD on intersecting lines.
STP=0 and b₁×b₂≠0 ⇒ intersecting coplanar.
Wrong a₁/a₂ plugs die when labels are written first.
Do thisWrite a₁,b₁,a₂,b₂ explicitly on rough sheet.
Prevents plugging the wrong point into SD.
SD numerator uses a₂−a₁ with those labels fixed.
Stops applying l²+m²+n²=1 to raw direction ratios.
Do thisIf identity needs l²+m²+n²=1, convert DRs to DCs first.
Raw DRs do not satisfy the cosine identity.
DRs ⟨2,3,6⟩ → divide by 7.
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
Find SD between r=i+λ(j−k) and r=2j+μ(i+k).
Gate skew first; then one SD formula.
Model chosenSD=|(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.
Answer√3
Takeaway: STP≠0 before quoting nonzero skew SD.
Write the vector equation of the line through A(1,0,0) and B(1,1,1).
b=B−A first, then r=A+λb.
Model chosenb=⟨0,1,1⟩; r=⟨1,0,0⟩+λ⟨0,1,1⟩.
Answerr = i + λ(j+k)
Takeaway: Two-point line = point + λ·displacement.
Find the shortest distance between the lines r = s⟨1, 1, 0⟩ and r = ⟨0, 1, 2⟩ + t⟨1, 0, 1⟩.
Skew SD is a template — 2D distance intuition fails here.
Model chosenSD = |(a₂ − a₁) · (b₁ × b₂)| / |b₁ × b₂| after non-coplanar gate.
Answer√3
Takeaway: Gate skew with STP ≠ 0 before quoting a nonzero SD.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likeEquation of line; angle between two lines
Fast modelExtract b; cosθ=|b₁·b₂|/(|b₁||b₂|).
All line angle work is just · of directions.
Do not mix up DRs with DC identity without converting.
Looks likeSD between two lines; show skew
Fast modelGate STP≠0; then SD formula.
Gate prevents wrong SD on intersecting lines.
Do not use plane–plane distance on skew lines.
Looks likeAre two lines coplanar? Find intersection
Fast model(a₂−a₁)·(b₁×b₂)=0 ⇒ coplanar; then solve λ,μ.
One scalar decides coplanar vs skew.
Do not call intersecting lines skew.
Looks likeFind line common to two planes; direction of intersection
Fast modelDirection b = n₁ × n₂; find one point by solving the system with a free variable.
Cross of normals is the direction — no parameter guessing.
Do not treat the intersection as the radical axis of two spheres unless asked that way.
Hall timing
Two-point form → b = P₂−P₁. Symmetric form → read l,m,n.
Parallel gate: b₁×b₂=0. Else prepare STP of (a₂−a₁),b₁,b₂.
Intersecting: solve λ,μ from a₁+λb₁=a₂+μb₂ when coplanar.
DCs satisfy l²+m²+n²=1 when they are cosines — DRs need not.
After the model is locked
(x−x₀)/l = (y−y₀)/m = (z−z₀)/n
Point + DRs
r = a + λb
Parametric / vector
(x−x₁)/(x₂−x₁)=…
Line through two points
cosθ = |b₁·b₂| / (|b₁||b₂|)
Direction vectors
|(a₂−a₁)·(b₁×b₂)| / |b₁×b₂|
Skew lines
(a₂−a₁)·(b₁×b₂)=0
Intersect or parallel in one plane
l=cosα, m=cosβ, n=cosγ; l²+m²+n²=1
Direction cosines
Divide DRs by √(l²+m²+n²)
Convert ratios to cosines
Make it automatic