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Math notesVectors & 3DTriple

Independent topic page

Scalar & Vector Triple Products

Triple-product full marks = STP as determinant for volume/coplanar, and BAC–CAB for a×(b×c) — never expand nested cross in raw components first.

Portion Vectors & 3DFocus Coplanarity, volume, nested-cross expansionsTarget 60–90sPortion share 11.2%Trend Rising
Official: scalar and vector triple products with geometrical interpretations. This page is only [a b c] and a×(b×c) — coplanarity, parallelepiped volume, and identity expansions. Triple products under the clock are determinant STP plus BAC–CAB — keep signs honest.

How Advanced asks this

  • Are three vectors coplanar?
  • Volume of parallelepiped
  • Simplify a×(b×c) or identity MC
  • Condition for four points coplanar (via vectors)
  • Joins with 3D coplanar lines (same STP idea)

Traps that leak marks

  • Forgetting | | for volume
  • Expanding nested cross in components first
  • Using position vectors instead of edge vectors for points
  • Wrong BAC–CAB order
  • Raw position vectors in STP instead of edge vectors from one vertex
  • Reporting signed det as volume without absolute value
  • Expanding nested cross component-wise when BAC–CAB applies

01. STP [a b c]=a·(b×c)=det[a b c]. Absolute value = parallelepiped volume. Zero ⇔ a,b,c coplanar (linearly dependent in 3D).

02. VTP a×(b×c)=(a·c)b−(a·b)c (BAC–CAB). Result lies in the plane of b and c.

03. Cyclic properties: [a b c]=[b c a]=[c a b]; swap two vectors flips sign.

04. Full marks: form the 3×3 for STP; write BAC–CAB before component chaos on nested cross.

05. Advanced lane: scalar triple product as a 3×3 determinant makes coplanarity and volume mechanical; write edge vectors from a common vertex for four-point checks.

06. Nested cross: BAC–CAB (or a×(b×c)=(a·c)b−(a·b)c) before component soup. Geometric volume is |[a b c]|, not the signed det alone.

Official · 2026

Syllabus for this topic

Official syllabus points for Scalar & Vector Triple Products only — not the full Vectors & 3D portion map. Portion hub: Vectors & 3D · Vectors & 3D analysis.

Syllabus on this page is topic-scoped: Triple products. The parent portion (Vectors & 3D) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Dot/cross/triple product geometry

Gaps / watch-outs

No topic-specific exclusions flagged — finish the unit points below for Triple.

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Triple products

medium

Exam focusScalar/vector triple products, volume, and coplanarity.

Syllabus points

  • Scalar and vector triple products and geometrical interpretations

Full Mathematics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Scalar & Vector Triple Products — practise the knowledge bridge, not only the single-chapter drill.

No tagged mixed stem matched this topic’s keywords yet. Still revise the joins listed under Trends, and scan the mixed joins board for Calculus / Coordinate / Algebra bridges.

All mixed joins →

Why methods matter

Common methods that change your score

Triple products are det + BAC–CAB. These methods make coplanar/volume and nested cross feel mechanical.

01

STP as determinant

Coplanarity and volume become a clean 3×3 — less sign chaos.

Do thisAlways compute [a b c] as a 3×3 det when components are given.

Why it works

Faster and fewer sign errors than expand-by-hand.

See it once

Coplanar ⇔ det=0 — one computation.

02

BAC–CAB for nested cross

Nested cross expansions stop exploding into component soup.

Do thisWrite a×(b×c)=(a·c)b−(a·b)c immediately.

Why it works

Keeps answer in span of b,c.

See it once

Never expand nested × in ijk first unless vectors are tiny numeric.

03

Volume = |STP|

Signed det is not the geometric volume — absolute value is.

Do thisTake absolute value after the determinant.

Why it works

Signed volume is not the geometric ask.

See it once

If det=−6, volume=6.

04

Points → edge vectors

Four-point coplanar checks stop using raw position vectors wrongly.

Do thisFor four points, form AB,AC,AD from A, then STP.

Why it works

Prevents using raw position vectors wrongly.

See it once

Coplanar ⇔ [AB AC AD]=0.

Processed examples

Sample questions · full working

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.

01

bankFamily · Coplanar / STP

Sample question

Show that a=i+j+k, b=2i+3j+4k, c=i+2j+3k are coplanar.

STP as det=0 is a one-line coplanar kill.

Model chosen[a b c]=det=0.

Processed solution

  1. Form the matrix with rows a,b,c: [[1,1,1],[2,3,4],[1,2,3]].
  2. det = 1(9−8) −1(6−4) +1(4−3) = 1 − 2 + 1 = 0.
  3. STP=0 ⇒ the three vectors are coplanar.

AnswerCoplanar (STP=0)

Takeaway: Coplanar ↔ [a b c]=0.

02

standardFamily · BAC–CAB

Sample question

Expand i×(j×k).

BAC–CAB before components — nested cross stays clean.

Model chosena×(b×c)=(a·c)b−(a·b)c with a=i,b=j,c=k.

Processed solution

  1. Apply BAC–CAB: a×(b×c)=(a·c)b−(a·b)c with a=i, b=j, c=k.
  2. i×(j×k)=(i·k)j−(i·j)k = 0·j − 0·k = 0.
  3. Check: j×k=i, so i×i=0 — matches.

Answer0

Takeaway: Write BAC–CAB first on every nested cross.

03

twistFamily · Coplanar / volume / STP

Sample question

For a = ⟨1, 1, 0⟩, b = ⟨0, 1, 1⟩, c = ⟨1, 0, λ⟩, find λ making them coplanar, then evaluate a × (b × c).

STP fixes the parameter; BAC–CAB finishes the nested cross — two families in one stem.

Model chosenSet [a b c] = 0; then a × (b × c) = (a · c)b − (a · b)c.

Processed solution

  1. b × c = ⟨λ, 1, −1⟩; a · (b × c) = λ + 1 = 0 ⇒ λ = −1.
  2. a · c = 1, a · b = 1.
  3. a × (b × c) = b − c = ⟨−1, 1, 2⟩ at λ = −1.

Answerλ = −1; a × (b × c) = ⟨−1, 1, 2⟩

Takeaway: Coplanarity first; BAC–CAB only after parameters are fixed.

Reusable models

Common question families

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.

01

Coplanar / volume / STP

Looks likeThree vectors coplanar? Parallelepiped volume?

Fast modeldet; coplanar iff 0; volume=|det|.

Same solve path

  1. 3×3 matrix.
  2. det.
  3. Interpret.

Rough skeleton

  • Matrix
  • det
  • Verdict

Why faster

Determinant beats expand-by-hand.

Refuse this

Do not forget | | for volume.

02

Nested cross / BAC–CAB

Looks likea×(b×c); simplify identity MC

Fast modela×(b×c)=(a·c)b−(a·b)c.

Same solve path

  1. Spot nested cross.
  2. BAC–CAB.
  3. Simplify dots.

Rough skeleton

  • BAC–CAB
  • Simplify

Why faster

Identity beats 9-component expansions.

Refuse this

Do not expand nested × in ijk first.

03

Four points coplanar

Looks likePoints A,B,C,D coplanar?

Fast model[AB AC AD]=0 using edge vectors from A.

Same solve path

  1. Form AB,AC,AD.
  2. STP.
  3. Zero?

Rough skeleton

  • Edges
  • det
  • 0?

Why faster

Edge vectors prevent position-vector mistakes.

Refuse this

Do not STP raw position vectors as if they were edges.

04

Reciprocal / cyclic STP identities

Looks like[a b c]=[b c a]; vanishing if two equal; linear dependence

Fast modelSTP is alternating; two equal args ⇒ 0; treat each MC claim as T/F.

Same solve path

  1. Use alternating property.
  2. Tick each option independently.
  3. Tiny basis test if needed.

Rough skeleton

  • Property
  • T/F

Why faster

Alternating rule kills half the options without computation.

Refuse this

Do not recompute a full det for a pure identity claim.

Hall timing

60–90s process

  1. 0–15s
    Name ask: coplanar / volume / expand VTP / identity T/F.

    Coplanar or volume → STP. Nested cross → BAC–CAB.

  2. 15–35s
    For STP, write 3×3 with components as rows (or columns) consistently.

    For VTP, write (a·c)b−(a·b)c before multiplying anything.

  3. 35–75s
    Compute det or simplify dots; interpret 0 vs |det|.

    Four points A,B,C,D coplanar ⇔ [AB,AC,AD]=0.

  4. 75–90s
    Check sign/permutation; volume uses absolute value.

    MC: swap test — swapping two vectors should flip STP sign.

After the model is locked

Quick formula bank

STP

[a b c]=a·(b×c)=det[a b c]

Volume / coplanar

Coplanar vectors

[a b c]=0

Three vectors coplanar

Volume

V = |[a b c]|

Parallelepiped spanned by a,b,c

VTP (BAC–CAB)

a×(b×c)=(a·c)b−(a·b)c

Nested cross expand

Cyclic STP

[a b c]=[b c a]=[c a b]

Reorder without sign change (cyclic)

Swap sign

[a c b]=−[a b c]

Odd permutation

Four points coplanar

[AB AC AD]=0

Points A,B,C,D

Mental bank

  • [a b c]=a·(b×c)=det
  • Coplanar ⇔ STP=0
  • Volume = |STP|
  • a×(b×c)=(a·c)b−(a·b)c
  • Cyclic STP equal; swap flips sign
  • Four points: [AB AC AD]=0
  • [a b c]=a·(b×c)=det with rows a,b,c
  • Coplanar a,b,c ⇔ [a b c]=0
  • a×(b×c)=(a·c)b−(a·b)c
  • Volume of parallelepiped = |[a b c]|

Make it automatic

Practice · checklist

Daily

  • 1 STP coplanar/volume.
  • 1 BAC–CAB expand.
  • Recite BAC–CAB once.

Weekly

  • 1 four-points coplanar.
  • 1 identity MC on STP properties.
  • Error log: missing absolute volume.

100% checklist

  • STP via determinant.
  • VTP via BAC–CAB first.
  • Volumes use |STP|.
  • Point problems use edge vectors.
  • ≥90% on last timed Triple set.
  • Four-point coplanar checks use edge vectors.
  • Volume answers use absolute STP.
  • Nested cross reduced by BAC–CAB before components.
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