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: 19 questions, with Q15 (section formula) and Q17 (right-angled triangle from position vectors) the two most paper-relevant items.
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 verified every magnitude, unit vector and section-formula result against the official NCERT key and the current 2026-27 print, with the internal-versus-external sign convention flagged on Q15 where most marks are lost.
Vector Algebra Class 12 Maths NCERT Solutions Exercise 10.2: Question-Wise Answer Map
The Vector Algebra Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 10.2 has 19 questions across five sub-topics. The table gives the final answer for each, so you can mark your own attempt without re-deriving.
| Q No. | Task | Answer |
|---|---|---|
| 1 | Magnitudes of three vectors | 3, √62, 1 |
| 2 | Two different vectors, same magnitude | i + j + k and i - j + k , both 3 |
| 3 | Two different vectors, same direction | i + j + k and 2i + 2j + 2k |
| 4 | Find x, y so vectors are equal | x = 2, y = 3 |
| 5 | Scalar and vector components, P(2,1) to Q(-5,7) | Scalars -7, 6; vectors -7i, 6j |
| 6 | Sum of three vectors | -4j - k |
| 7 | Unit vector along i + j + 2k | i + j + 2k6 |
| 8 | Unit vector along PQ , P(1,2,3), Q(4,5,6) | i + j + k3 |
| 9 | Unit vector along a + b | i + k2 |
| 10 | Vector of magnitude 8 along 5i - j + 2k | 8(5i - j + 2k)√30 |
| 11 | Show 2i - 3j + 4k and -4i + 6j - 8k collinear | b = -2a , so collinear |
| 12 | Direction cosines of i + 2j + 3k | (1√14, 2√14, 3√14) |
| 13 | Direction cosines of AB , A(1,2,-3), B(-1,-2,1) | (-13, -23, 23) |
| 14 | Show i + j + k equally inclined to axes | l = m = n = 13 |
| 15 | Position vector of R dividing PQ in 2:1 | Internal (-13, 43, 13) ; external (-3, 0, 3) |
| 16 | Midpoint of P(2,3,4) and Q(4,1,-2) | 3i + 2j + k |
| 17 | Show A, B, C form a right-angled triangle | |AB|2 + |AC|2 = |BC|2 , right angle at A |
| 18 | MCQ: which is not true in triangle ABC | Option (C) |
| 19 | MCQ: incorrect statements on collinear vectors | Options (B), (C), (D) |
Q15 is the single most paper-relevant item: the internal and external section formulae differ only by the sign before np , and that is where most candidates drop marks. A short-answer on the section formula or on a unit vector has appeared in every one of the last 5 CBSE Class 12 board papers.
Vector Algebra Ex 10 2 Video Walkthrough
Source: Magnet Brains on YouTube
Core Formulae Behind Class 12 Maths Chapter 10 Exercise 10.2
These notes address this in the same order as the NCERT textbook.
Almost every question in this exercise is one of five formulae applied once. Keeping them in this order matches the order the questions appear in.
Unit vector: a = a|a|
Direction cosines: l = a1|a|, m = a2|a|, n = a3|a| with l2 + m2 + n2 = 1
Section formula (m:n): internal mq + npm+n , external mq - npm-n
Collinearity: b = λ a , i.e. components proportional
The check l2 + m2 + n2 = 1 is the fastest self-test for Q12 to Q14: if the squared direction cosines do not add to 1, a component or the magnitude is wrong.
Concept Tags Across the 19 Problems of Class 12 Maths Exercise 10.2
The this Class 12 page address this in the same order as the NCERT textbook.
The exercise is best practised sub-topic by sub-topic rather than in serial order. The grouping below shows which formula each question needs.
| Sub-topic | Questions |
|---|---|
| Magnitude and equal vectors | Q1, Q2, Q4 |
| Same-direction and component decomposition | Q3, Q5, Q6 |
| Unit vectors | Q7, Q8, Q9, Q10 |
| Collinearity | Q11, Q19 |
| Direction cosines | Q12, Q13, Q14 |
| Section formula and midpoint | Q15, Q16 |
| Triangle geometry and triangle law | Q17, Q18 |
Direction-cosine problems (Q12 to Q14) and the section formula (Q15) are the two clusters that recur most in CBSE short answers, so practise those two groups first.
How Collegedunia's NCERT Solutions for Vector Algebra Help You in Exercise 10.2
Exercise 10.2 is mechanical once the right formula is picked, and the marks are lost on sign and order, not on the idea. The Collegedunia solutions state the formula, then substitute on a separate line, then do the arithmetic, so each step is independently checkable.
- Section-formula sign flagged: Q15 shows internal with +np and external with -np side by side, with a midpoint sanity check.
- Magnitude before unit vector: Q7 to Q10 always compute |a| on its own line before dividing.
- Direction-cosine self-test: every Q12 to Q14 answer ends with l2 + m2 + n2 = 1 verified.
- Pythagoras converse for Q17: squared side-lengths are compared, avoiding surds and locating the right angle precisely.
Common Mistakes Students Make in Class 12 Maths Exercise 10.2
The the resource are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Dropping the minus sign in the external section formula and getting the internal point instead.
- Forgetting to divide by |a| when "unit vector" is asked, leaving an un-normalised vector.
- Confusing scalar components (numbers) with vector components (numbers times i, j, k ) in Q5.
- Reading the wrong vertex for the right angle in Q17; the right angle is opposite the longest side.
- Calling two vectors collinear from one matching ratio; all component ratios must be equal.
Vector Algebra Class 12 Maths NCERT Solutions Exercise 10.2: 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 10.3 | Exercise 10.3 Solutions (Scalar or Dot Product) |
| 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 10.2 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 10 Vector Algebra Ex 10.2 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
Compute the magnitude of the following vectors:
a = i + j + k; b = 2i - 7j - 3k; c = 13i + 13j - 13k.
Write two different vectors having same magnitude.
Write two different vectors having same direction.
Find the values of x and y so that the vectors 2i + 3j and xi + yj are equal.
Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (-5, 7).
Find the sum of the vectors a = i - 2j + k, b = -2i + 4j + 5k and c = i - 6j - 7k.
Find the unit vector in the direction of the vector a = i + j + 2k.
Find the unit vector in the direction of vector PQ, where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.
For given vectors a = 2i - j + 2k and b = -i + j - k, find the unit vector in the direction of the vector a + b.
Find a vector in the direction of vector 5i - j + 2k which has magnitude 8 units.
Show that the vectors 2i - 3j + 4k and -4i + 6j - 8k are collinear.
Find the direction cosines of the vector i + 2j + 3k.
Find the direction cosines of the vector joining the points A(1, 2, -3) and B(-1, -2, 1), directed from A to B.
Show that the vector i + j + k is equally inclined to the axes OX, OY and OZ.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are i + 2j - k and -i + j + k respectively, in the ratio 2:1
(i) internally (ii) externally.
Find the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).
Show that the points A, B and C with position vectors a = 3i - 4j - 4k, b = 2i - j + k and c = i - 3j - 5k, respectively form the vertices of a right-angled triangle.
In triangle ABC (Fig 10.18), which of the following is not true:
(A) AB + BC + CA = 0
(B) AB + BC - AC = 0
(C) AB + BC - CA = 0
(D) AB - CB + CA = 0
If a and b are two collinear vectors, then which of the following are incorrect:
(A) b = λ a, for some scalar λ
(B) a = ± b
(C) the respective components of a and b are not proportional
(D) both the vectors a and b have same direction, but different magnitudes.
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 10.2?
Ans. Exercise 10.2 of Class 12 Maths Chapter 10 Vector Algebra has 19 questions in the 2026-27 NCERT. Seventeen are short-answer problems on magnitude, unit vectors, components, direction cosines and the section formula, and the last two (Q18, Q19) are single-correct MCQs.
Ques. What is the section formula in Class 12 Maths Chapter 10 Exercise 10.2?
Ans. If R divides the segment PQ in the ratio m:n, the internal division point is mq + npm+n and the external division point is mq - npm-n . Both are used in Q15, where the only difference is the sign before np .
Ques. How do you find the unit vector in the direction of a vector in Exercise 10.2?
Ans. Divide the vector by its magnitude: a = a|a| . For example in Q7, |i + j + 2k| = 6 , so the unit vector is i + j + 2k6 . A correct unit vector always satisfies |a| = 1 .
Ques. How do you show two vectors are collinear in Class 12 Maths Chapter 10?
Ans. Two vectors are collinear when one is a scalar multiple of the other, b = λ a , which means all corresponding components are in the same ratio. In Q11, -4i + 6j - 8k = -2(2i - 3j + 4k) , so the two vectors are collinear with λ = -2 .
Ques. How do I download the Class 12 Maths Chapter 10 Exercise 10.2 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 10.2 NCERT Solutions PDF. The file is free, ad-free and mapped to the 2026-27 NCERT edition.
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