This page hosts the Vector Algebra Class 12 Formulas, and hosts the Vector Algebra Class 12 Formulas for Class 12 Mathematics Chapter 10 Vector Algebra. The file presents the formula sheet in the order they appear in the NCERT textbook, with each entry followed by a one-line note on its standard use. the PDF are intended for last-revision and exam-day recall.
- CBSE Weightage: 5 to 7 marks, typically one 5-mark long answer on cross product or projection plus a 2-mark short answer on direction cosines.
The full sheet carries 38 working formulas across 6 NCERT sections, organised by operation so a student can map a question stem to a recipe in under thirty seconds.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
Vector Algebra Class 12 Formulas: Sheet Structure
These notes sheet compresses 40 NCERT pages into six pages of pure formulas, organised by operation so the student can map a question stem to a recipe in seconds. Roughly 75% of CBSE board marks on Vector Algebra are awarded for correctly applying the dot product, cross product, and projection formulas verbatim.
- Operation-first layout groups formulas by vector operation (addition, dot, cross, projection), not by NCERT section.
- Cross product block isolates the determinant form, the area formulas, and the right-hand rule as a standalone box.
- PYQ hit-rate tags flag which standard forms appeared in CBSE and JEE Main 2021 to 2025 papers.
Vector Algebra Video Walkthrough
Source: Magnet Brains on YouTube
Position Vector and Section Formula for Class 12 Vector Algebra
The this Class 12 page address this in the same order as the NCERT textbook.
NCERT Section 10.4 defines the position vector and the section formula, both of which feed directly into 3D Geometry in Chapter 11. Most multi-part board questions on vectors open with a position-vector setup.
| Concept | Formula | Use Case |
|---|---|---|
| Position Vector | OP⃗ = xî + yĵ + zk̂ for point P(x, y, z) | Locate any point in 3D space relative to the origin. |
| Magnitude | OP⃗ = √x2 + y2 + z2 | Length of position vector from origin to P. |
| Vector Joining Two Points | AB⃗ = OB⃗ - OA⃗ = (x2-x1)î + (y2-y1)ĵ + (z2-z1)k̂ | Direction vector from A to B. |
| Internal Section Formula | r⃗ = mb⃗ + na⃗m + n | Point dividing segment AB internally in ratio m:n. |
| External Section Formula | r⃗ = mb⃗ - na⃗m - n | Point dividing segment AB externally in ratio m:n. |
| Midpoint | r⃗ = a⃗ + b⃗2 | Midpoint of segment joining A and B. |
Unit Vector, Direction Cosines and Direction Ratios for Class 12 Maths
The the resource address this in the same order as the NCERT textbook.
Direction cosines and direction ratios are a fixed 2-mark short answer in CBSE. Direction cosines are the cosines of the angles a vector makes with the x, y and z axes; direction ratios are any three numbers proportional to those cosines.
| Concept | Formula | Quick Note |
|---|---|---|
| Unit Vector | â = a⃗a⃗ | Direction of a⃗ with unit length. |
| Standard Basis | î, ĵ, k̂ along x, y, z axes | Each has magnitude 1; mutually perpendicular. |
| Direction Cosines | l = cosα = a1a⃗ , m = cosβ = a2a⃗ , n = cosγ = a3a⃗ | For a⃗ = a1î + a2ĵ + a3k̂. |
| DC Identity | l2 + m2 + n2 = 1 | Always true for direction cosines. |
| Direction Ratios | a : b : c = l : m : n | Any non-zero scalar multiple of the DCs. |
| DCs from DRs | l = a√a2+b2+c2 and similar for m, n | Normalise the direction ratios. |
Scalar (Dot) Product Class 12 Maths Formula Sheet
The chapter notes address this in the same order as the NCERT textbook.
The scalar product is the workhorse of Vector Algebra. Every CBSE Class 12 Maths paper since 2019 has carried at least one 2-mark or 3-mark question on the dot product. Two forms must be memorised: trigonometric for angle problems, algebraic for component vectors.
| Form | Formula | Use Case |
|---|---|---|
| Trigonometric | a⃗ · b⃗ = a⃗ b⃗ cosθ | When magnitudes and angle are given. |
| Algebraic (Components) | a⃗ · b⃗ = a1b1 + a2b2 + a3b3 | When component forms are given. |
| Angle Between Vectors | cosθ = a⃗ · b⃗a⃗ b⃗ | Rearrangement to find θ. |
| Perpendicularity Test | a⃗ · b⃗ = 0 a⃗ ⊥ b⃗ (both non-zero) | Quick check from components. |
| Projection of a⃗ on b⃗ | projb⃗a⃗ = a⃗ · b⃗b⃗ | Scalar component along b⃗. |
| Vector Projection | p⃗ = (a⃗ · b⃗b⃗ 2)b⃗ | Vector form of the projection. |
| Square of Magnitude | a⃗ · a⃗ = a⃗ 2 | Self-dot equals magnitude squared. |
| Distributivity | a⃗ · (b⃗ + c⃗) = a⃗ · b⃗ + a⃗ · c⃗ | Linearity over addition. |
Vector (Cross) Product Formula Sheet for Class 12 Maths Chapter 10
Cross product is the highest-marks topic in the the PDF. The 5-mark long-answer slot in CBSE Class 12 Vector Algebra is almost always built around it, either to compute an area or to find a vector perpendicular to two given vectors.
| Form | Formula | Use Case |
|---|---|---|
| Trigonometric | a⃗ × b⃗ = a⃗ b⃗ sinθ n̂ | n̂ is the unit vector by right-hand rule. |
| Determinant Form | a⃗ × b⃗ = vmatrix î & ĵ & k̂ a1 & a2 & a3 b1 & b2 & b3 vmatrix | Default method for component vectors. |
| Magnitude | a⃗ × b⃗ = a⃗ b⃗ sinθ | θ is the angle between vectors. |
| Parallelism Test | a⃗ × b⃗ = 0⃗ a⃗ ∥ b⃗ (both non-zero) | Quick check from components. |
| Area of Parallelogram (adjacent sides) | A = a⃗ × b⃗ | Sides a⃗ and b⃗ from one vertex. |
| Area of Parallelogram (diagonals) | A = 12d1⃗ × d2⃗ | Diagonals d1⃗ and d2⃗. |
| Area of Triangle | A = 12AB⃗ × AC⃗ | Triangle with vertices A, B, C. |
| Anti-commutative | a⃗ × b⃗ = -(b⃗ × a⃗) | Cross product is sign-sensitive to order. |
| Unit Vector Perpendicular to Both | n̂ = a⃗ × b⃗a⃗ × b⃗ | Direction normal to the plane of a⃗, b⃗. |
Scalar Triple Product and Standard Vector Identities in Class 12 Maths
The scalar triple product appears as a 2-mark or 3-mark coplanarity question in most CBSE papers. The identities below cover the rest of the short-answer scoring.
| Identity | Formula | Where Tested |
|---|---|---|
| Scalar Triple Product | [a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗) = vmatrix a1 & a2 & a3 b1 & b2 & b3 c1 & c2 & c3 vmatrix | Volume of parallelepiped; coplanarity check. |
| Coplanarity Condition | [a⃗ b⃗ c⃗] = 0 iff a⃗, b⃗, c⃗ are coplanar. | CBSE 2024, 2022 |
| Lagrange's Identity | a⃗ × b⃗ 2 + (a⃗ · b⃗)2 = a⃗ 2 b⃗ 2 | JEE Main 2024, CBSE 2023 |
| Three Collinear Points | AB⃗ = λ AC⃗ for some scalar λ | JEE Main 2023, CUET 2024 |
Top 5 PYQ Templates Class 12 Vector Algebra Sets Every Year
Across CBSE Class 12 Maths papers 2021 to 2025, Vector Algebra questions cluster into five repeating templates. Drilling these covers more than 90% of past board-paper variants.
| PYQ Template | Formula Used | Last Asked |
|---|---|---|
| Find a unit vector perpendicular to two given vectors | n̂ = a⃗ × b⃗a⃗ × b⃗ | CBSE 2024, JEE Main 2025 |
| Area of triangle with given vertices | A = 12AB⃗ × AC⃗ | CBSE 2025 |
| Angle between two vectors | cosθ = a⃗ · b⃗a⃗ b⃗ | CBSE 2023, JEE Main 2024 |
| Projection of one vector on another | projb⃗a⃗ = a⃗ · b⃗b⃗ | CBSE 2022, CUET 2024 |
| Direction cosines from two given points | l = x2-x1√∑ (x2-x1)2 and similar | CBSE 2021, 2025 |
Full year-wise PYQ map: See the dedicated Vector Algebra NCERT Solutions for Class 12 Maths for fully solved PYQs with marking-scheme step weightages.
Vector Algebra Chapter Weightage Across CBSE, JEE Main and CUET
The marks share of Vector Algebra across the three Class 12 exams is summarised below. JEE Main 2025 led the year with two questions tagged to the this chapter; CBSE Board 2025 carried a 5-mark long answer plus a 2-mark short answer.
| Exam | 2025 | 2024 | 2023 | 2022 | 2021 |
|---|---|---|---|---|---|
| CBSE Class 12 Maths | 7 marks | 5 marks | 6 marks | 5 marks | 6 marks |
| JEE Main (Maths) | 2 Qs | 3 Qs | 2 Qs | 2 Qs | 1 Q |
| CUET (UG) Maths | 2 Qs | 3 Qs | 2 Qs | 1 Q | 1 Q |
Common Mistakes in Vector Algebra Class 12 Maths
These notes are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
The errors below cost Class 12 students 1 to 3 marks per Vector Algebra question.
- Confusing dot with cross product: dot returns a scalar, cross returns a vector. Writing a⃗ · b⃗ as a vector loses 1 mark.
- Forgetting the sinθ in the magnitude of the cross product.
- Skipping normalisation when asked for direction cosines (returning direction ratios instead).
- Reversing the cross-product order without flipping the sign.
- Using the dot product for the area of a parallelogram (it is the magnitude of the cross product).
Related Resources for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Notes for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra
- Class 12 Maths Vector Algebra Handwritten Notes
- NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Book PDF for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Exemplar Book PDF for Class 12 Maths Chapter 10 Vector Algebra
NCERT Formula Sheet for Class 12 Maths: All Chapters
Quick links for cross-chapter revision after Vector Algebra and before 3D Geometry.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
this Class 12 page: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Vector Algebra Chapter
The Vector Algebra chapter splits into 4 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 10.1 | Vectors and scalars; direction cosines and ratios |
| Exercise 10.2 | Algebra of vectors; section formula |
| Exercise 10.3 | Scalar (dot) product of vectors |
| Exercise 10.4 | Vector (cross) product of vectors |
| Miscellaneous Exercise | Mixed vector algebra problems |
PDF Download Formats and Languages for the Vector Algebra Chapter
The Vector Algebra Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The vector algebra class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the vector algebra class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The vector algebra class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The chapter notes pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The vector algebra class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the the PDF pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Vector Algebra Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Vector Algebra chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable vector algebra class 12 important questions you will see on board day.
- vector algebra class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The the resource important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching the chapter notes extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Vector Algebra chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full the PDF important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Vector Algebra Notes Pair with NCERT Solutions and the Formula Sheet
The this chapter notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Vector Algebra Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| vector algebra class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| this chapter PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The vector algebra class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The vector algebra class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The these notes reference sheet is the same A4 file students sometimes refer to as these notes all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board this Class 12 page textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 vector algebra solutions, class 12 vector algebra ncert solutions, ncert class 12 vector algebra solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to the resource |
|---|---|
| RD Sharma Class 12 Vector Algebra | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Vector Algebra | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the chapter notes | Free online walkthroughs; useful for video-style learning. |
| Shaalaa vector algebra class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board the PDF textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Vector Algebra | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Vector Algebra Notes Page Most Effectively
The recommended study plan for the this chapter chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the vector algebra class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The these notes important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Comments