The Vector Algebra Class 12 NCERT Solutions page compiles NCERT Class 12 Mathematics Chapter 10 into a download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, every formula matched to the CBSE marking scheme.
The solutions PDF covers all parts of every question and writes each step in full. The file is part of the NCERT Solutions Class 12 Maths library and follow the notation of the official textbook.
- Question count: 18 questions, with 17 short and proof-type problems plus one single-correct MCQ (Q18).
Student Pulse - Vector Algebra Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
The Collegedunia editorial team has checked every angle, projection and identity against the official NCERT key and the current 2026-27 print, with the difference-of-squares identity (used in Q6, Q9, Q11) flagged because it collapses several questions to a single line.
Why the Scalar Product Carries Weight in Class 12 Maths Chapter 10 Exercise 10.3
The Vector Algebra Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 10.3 is the highest-yield numbered set in Chapter 10 for the board paper, because the dot product is the only tool needed for the angle between vectors, the projection of one vector on another, and the perpendicularity test, three of the most repeated short-answer stems. Getting the single formula right here pays off directly in Exercise 10.4 and the Miscellaneous set.
A short-answer on the angle between two vectors or on a projection has appeared in 5 of the last 5 CBSE Class 12 board papers. The proof-type Q11 (sum and difference of equal-magnitude vectors are perpendicular) is also a frequent 3-mark item.
Vector Algebra Ex 10 3 Video Walkthrough
Source: Magnet Brains on YouTube
How Collegedunia's NCERT Solutions for Exercise 10.3 Help You Score
These notes address this in the same order as the NCERT textbook.
The dot product is forgiving on ideas but unforgiving on sign and on which vector goes in the denominator of a projection. The Collegedunia solutions separate the formula, the substitution and the arithmetic onto their own lines so a marker can follow each step, which is exactly how CBSE awards method marks.
- Identity flagged early: Q6, Q9 and Q11 all use (u + v)·(u - v) = |u|2 - |v|2 , called out so you reach for it instead of expanding components.
- Projection denominator fixed: projection of a on b is a · b|b| , with the dividing vector stated explicitly in Q3 and Q4.
- Angle left exact: answers such as cos-1(5/7) are kept exact, matching the NCERT key, not rounded to degrees.
- Zero-dot reasoning: Q12 and Q14 separate "a vector is zero" from "the vectors are perpendicular", the two ways a dot product vanishes.
Vector Algebra Class 12 Maths NCERT Solutions Exercise 10.3: Question-Wise Answer Map
The this Class 12 page address this in the same order as the NCERT textbook.
Exercise 10.3 has 18 questions on the scalar product. The table records the final answer for each, so you can verify your working quickly.
| Q No. | Task | Answer |
|---|---|---|
| 1 | Angle, given |a|=3, |b|=2, a·b=6 | θ = π/4 |
| 2 | Angle between i - 2j + 3k and 3i - 2j + k | cos-1(5/7) |
| 3 | Projection of i - j on i + j | 0 (perpendicular) |
| 4 | Projection of i + 3j + 7k on 7i - j + 8k | 60√114 |
| 5 | Show three given vectors are mutually perpendicular unit vectors | Each magnitude 1, all pairwise dot products 0 |
| 6 | Find |a|, |b| from the given conditions | |a| = 16237, |b| = 2237 |
| 7 | Evaluate (3a - 5b)·(2a + 7b) | 6|a|2 + 11(a·b) - 35|b|2 |
| 8 | Magnitude of equal-length vectors, angle 60 degrees, dot 12 | |a| = |b| = 1 |
| 9 | Find |x| from (x - a)·(x + a) = 12 , a unit | √13 |
| 10 | Find λ so a + λb ⊥ c | λ = 8 |
| 11 | Show |a|b + |b|a ⊥ |a|b - |b|a | Dot product 0, hence perpendicular |
| 12 | Conclusion from a·a = 0 and a·b = 0 | a = 0 , so b is any vector |
| 13 | Value of a·b + b·c + c·a , unit vectors summing to zero | -32 |
| 14 | Converse of "a or b zero implies dot zero" | False; counter-example i, j |
| 15 | Angle ABC for A(1,2,3), B(-1,0,0), C(0,1,2) | cos-1(10√102) |
| 16 | Show A(1,2,7), B(2,6,3), C(3,10,-1) collinear | AB = BC , so collinear |
| 17 | Show three vectors form a right-angled triangle | |BC|2 + |CA|2 = |AB|2 , right angle at C |
| 18 | MCQ: λa is a unit vector if | Option (D), a = 1/|λ| |
Q3 returns a projection of 0, which is the cleanest signal that the two vectors are perpendicular. The difference-of-squares identity does the heavy lifting in Q6, Q9 and Q11, turning each into a one-line equation.
Dot-Product Toolkit for Class 12 Maths Exercise 10.3
Five formulae cover the whole exercise. Reading them in this order matches how the questions are arranged.
Angle: cosθ = a·b|a||b|
Projection of a on b : a·b|b|
Perpendicular: a·b = 0 (for non-zero vectors)
Workhorse identity: (u + v)·(u - v) = |u|2 - |v|2
The last identity is the one to spot first: it converts Q6, Q9 and Q11 from component grinding into a single algebraic equation in magnitudes.
Concept Tags Across the 18 Problems of Class 12 Maths Exercise 10.3
Practising by sub-topic is faster than serial order. The grouping shows which tool each question needs.
| Sub-topic | Questions |
|---|---|
| Angle between vectors | Q1, Q2, Q15 |
| Projection | Q3, Q4 |
| Perpendicularity and orthonormal sets | Q5, Q10, Q11 |
| Difference-of-squares identity | Q6, Q9 |
| Dot-product algebra and zero-dot reasoning | Q7, Q12, Q13, Q14 |
| Geometry: equal magnitude, collinear, right triangle | Q8, Q16, Q17 |
The angle cluster (Q1, Q2, Q15) and the projection pair (Q3, Q4) are the two groups CBSE reuses most often in short answers, so drill them before the proof-type questions.
Common Mistakes Students Make in Class 12 Maths Exercise 10.3
The the resource are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Rounding cos-1(5/7) to a degree value; the NCERT key keeps the exact inverse cosine.
- Treating a·b = 0 as proof that a vector is zero; it can also mean the vectors are perpendicular (Q12, Q14).
- Expanding components in Q6, Q9 and Q11 instead of using the difference-of-squares identity.
- Forgetting that the dot product is commutative when collecting like terms in Q7.
- Using the wrong pair of edge-vectors for the angle at a vertex in Q15; both must point away from that vertex.
Vector Algebra Class 12 Maths NCERT Solutions Exercise 10.3: Related Exercises
The table lists the other exercises of Class 12 Maths Chapter 10 currently published, each mapped to the 2026-27 NCERT.
| Exercise | NCERT Solutions |
|---|---|
| Exercise 10.1 | Exercise 10.1 Solutions (Scalars, Vectors, Types of Vectors) |
| Exercise 10.2 | Exercise 10.2 Solutions (Magnitude, Unit Vectors, Section Formula) |
| Exercise 10.4 | Exercise 10.4 Solutions (Vector or Cross Product) |
| Miscellaneous | Miscellaneous Exercise Solutions (Mixed Vector Problems) |
Related Resources for Class 12 Maths Chapter 10 Vector Algebra
- Formula Sheet for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Notes for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Book PDF for Class 12 Maths Chapter 10 Vector Algebra
- NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra
the chapter notes: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
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 vector algebra class 12 formulas 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 vector algebra class 12 formulas 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 vector algebra class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching this chapter 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 vector algebra class 12 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
These notes 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 |
| vector algebra class 12 formulas 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 this chapter 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 vector algebra class 12 formulas reference sheet is the same A4 file students sometimes refer to as this Class 12 page all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource 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 chapter notes |
|---|---|
| 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 PDF | Free online walkthroughs; useful for video-style learning. |
| Shaalaa vector algebra class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board this chapter 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 these notes 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 these notes 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 vector algebra class 12 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.
All NCERT Solutions for Vector Algebra Ex 10.3 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 10 Vector Algebra Ex 10.3 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Find the angle between two vectors a and b with magnitudes 3 and 2, respectively having a·b = 6.
Find the angle between the vectors i - 2j + 3k and 3i - 2j + k.
Find the projection of the vector i - j on the vector i + j.
Find the projection of the vector i + 3j + 7k on the vector 7i - j + 8k.
Show that each of the given three vectors is a unit vector: 17(2i + 3j + 6k), 17(3i - 6j + 2k), 17(6i + 2j - 3k). Also, show that they are mutually perpendicular to each other.
Find |a| and |b|, if (a + b)·(a - b) = 8 and |a| = 8|b|.
Evaluate the product (3a - 5b)·(2a + 7b).
Find the magnitude of two vectors a and b, having the same magnitude and such that the angle between them is 60∘ and their scalar product is 12.
Find |x|, if for a unit vector a, (x - a)·(x + a) = 12.
If a = 2i + 2j + 3k, b = -i + 2j + k and c = 3i + j are such that a + λb is perpendicular to c, then find the value of λ.
Show that |a|b + |b|a is perpendicular to |a|b - |b|a, for any two non-zero vectors a and b.
If a·a = 0 and a·b = 0, then what can be concluded about the vector b?
If a, b, c are unit vectors such that a + b + c = 0, find the value of a·b + b·c + c·a.
If either vector a = 0 or b = 0, then a·b = 0. But the converse need not be true. Justify your answer with an example.
If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1, 2) respectively, then find ∠ ABC. [∠ ABC is the angle between the vectors BA and BC.]
Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.
Show that the vectors 2i - j + k, i - 3j - 5k and 3i - 4j - 4k form the vertices of a right-angled triangle.
If a is a non-zero vector of magnitude `a' and λ a non-zero scalar, then λ a is unit vector if
(A) λ = 1 (B) λ = -1 (C) a = |λ| (D) a = 1/|λ|.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Vector Algebra Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are in Class 12 Maths Chapter 10 Exercise 10.3?
Ans. Exercise 10.3 of Class 12 Maths Chapter 10 Vector Algebra has 18 questions in the 2026-27 NCERT. Seventeen are short-answer and proof-type problems on the scalar product, angle, projection and perpendicularity, and Q18 is a single-correct MCQ.
Ques. How do you find the angle between two vectors in Class 12 Maths Chapter 10?
Ans. Use cosθ = a·b|a||b| , then take the inverse cosine. In Q2, a·b = 10 and |a| = |b| = √14 , so cosθ = 5/7 and θ = cos-1(5/7) .
Ques. What is the projection of a vector on another in Exercise 10.3?
Ans. The projection of a on b is a·b|b| , the signed length of the shadow of a along b . In Q3 the projection of i - j on i + j is 0, which shows the two vectors are perpendicular.
Ques. If three unit vectors sum to zero, what is the sum of their pairwise dot products?
Ans. Squaring a + b + c = 0 gives 3 + 2(a·b + b·c + c·a) = 0 , so the sum equals -32 . This is Q13, and geometrically the three unit vectors lie at 120 degrees to each other.
Ques. How do I download the Class 12 Maths Chapter 10 Exercise 10.3 NCERT Solutions PDF?
Ans. Use the green download button on the this chapter card at the top of these notes to save the Collegedunia Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 NCERT Solutions PDF. The file is free, ad-free and mapped to the 2026-27 NCERT edition.
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