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 15 short and long-answer problems and 4 single-correct MCQs (Q16 to Q19).
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 answer against the official NCERT key and the current 2026-27 print, with the multi-operation questions (Q12 perpendicular vector with a dot constraint, Q9 collinearity plus ratio) annotated so the order of operations is unambiguous.
Vector Algebra Class 12 Maths NCERT Solutions Miscellaneous Exercise: Question-Wise Answer Map
The Vector Algebra Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The Miscellaneous Exercise has 19 questions spanning every Chapter 10 tool. The table records the final answer for each, so you can verify a long attempt without re-deriving.
| Q No. | Task | Answer |
|---|---|---|
| 1 | Unit vector in XY-plane, 30 degrees from positive x-axis | 32i + 12j |
| 2 | Scalar components and magnitude of PQ | Components (x2-x1, y2-y1, z2-z1) ; magnitude the root of their squares |
| 3 | Girl's displacement: 4 km west, then 3 km, 30 degrees east of north | -52i + 332j , magnitude √13 km |
| 4 | Is |a| = |b| + |c| when a = b + c ? | No; equality only if b, c are like-parallel |
| 5 | Value of x so x(i + j + k) is a unit vector | x = ±13 |
| 6 | Vector of magnitude 5, parallel to a + b | 5√10(3i + j) |
| 7 | Unit vector parallel to 2a - b + 3c | 1√22(3i - 3j + 2k) |
| 8 | Show A, B, C collinear; ratio in which B divides AC | Collinear; B divides AC internally 2:3 |
| 9 | Position vector of R dividing PQ externally 1:2 | r = 3a + 5b ; P is midpoint of RQ |
| 10 | Unit vector along the diagonal and area of a parallelogram | 17(3i - 6j + 2k) ; area 115 |
| 11 | Show direction cosines of an equally inclined vector are ±(1/3,) | (l,m,n) = ±(13, 13, 13) |
| 12 | Vector d ⊥ a, b with c·d = 15 | d = 13(160i - 5j - 70k) |
| 13 | Find λ from a scalar-product equal to one condition | λ = 1 |
| 14 | Show a + b + c equally inclined to mutually perpendicular equal vectors | Equal angle cos-1(1/3) with each |
| 15 | Prove (a+b)·(a+b) = |a|2 + |b|2 a⊥b | Equivalent to a⊥b |
| 16 | MCQ: a·b ≥ 0 only when | Option (B), 0 ≤ θ ≤ π/2 |
| 17 | MCQ: a + b a unit vector if | Option (D), θ = 2π/3 |
| 18 | MCQ: value of i·(j×k) + j·(i×k) + k·(i×j) | Option (C), value 1 |
| 19 | MCQ: |a·b| = |a×b| when θ is | Option (B), θ = π/4 |
Q12 is the signature mixed question: it requires a cross product to get a vector perpendicular to both a and b , then a dot-product constraint to fix the scale. A mixed Miscellaneous-style problem has seeded the 5-mark Vector Algebra long answer in 4 of the last 5 CBSE board papers.
Vector Algebra Misc Video Walkthrough
Source: Magnet Brains on YouTube
Tools Combined Across Class 12 Maths Chapter 10 Miscellaneous Exercise
The this chapter address this in the same order as the NCERT textbook.
Almost every Miscellaneous question chains two or three operations. The reference box lists the ones that recur, in the order they typically appear within a problem.
Section formula: internal mq + npm+n , external mq - npm-n
Perpendicular vector: a×b is perpendicular to both a and b
Dot constraint: use c·d = k to fix the unknown scalar
Equally inclined: direction cosines ±(1/3, 1/3, 1/3)
Q12 walks the full chain: cross product for the direction, then a dot-product equation for the scale, which is the exact structure of these notes's 5-mark question.
Concept Tags Across the 19 Problems of Class 12 Maths Miscellaneous Exercise
The this Class 12 page address this in the same order as the NCERT textbook.
Sorting the questions by the dominant tool shows which clusters carry the most board weight.
| Sub-topic | Questions |
|---|---|
| Unit and parallel vectors | Q1, Q5, Q6, Q7 |
| Displacement and triangle inequality | Q3, Q4 |
| Collinearity and section formula | Q8, Q9 |
| Cross product, area, perpendicular vector | Q10, Q12 |
| Equally inclined vectors | Q11, Q14 |
| Dot-product proofs and MCQs | Q13, Q15, Q16, Q17, Q18, Q19 |
The cross-product cluster (Q10, Q12) and the section-formula pair (Q8, Q9) are the two groups that most often become the long answer, so practise those before the proof-type and MCQ items.
How Collegedunia's NCERT Solutions for the Miscellaneous Exercise Help You
The Miscellaneous set is hard not because any single step is difficult but because the steps must be chained in the right order. The Collegedunia solutions number each operation, show the formula then the substitution then the arithmetic on separate lines, and state the constraint that fixes the final scalar.
- Order of operations: Q12 does the cross product first, then applies c·d = 15 to fix the multiplier.
- Triangle inequality reasoning: Q4 explains that |a| = |b| + |c| holds only when b and c point the same way.
- Section formula sign: Q9 uses external division and then proves P is the midpoint of RQ as a built-in check.
- Triple product MCQ: Q18 evaluates each scalar triple product against the i, j, k cyclic rule, landing on value 1.
Common Mistakes Students Make in Class 12 Maths Miscellaneous Exercise
The the resource are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Writing |a| = |b| + |c| as always true; it holds only for like-parallel b, c (Q4).
- Using the internal section formula where Q9 asks for external division.
- Forgetting the ± on equally inclined direction cosines in Q11 and Q14.
- Mishandling the bearing in Q3; "30 degrees east of north" is measured from the north ray.
- Treating a scalar triple product as a vector in Q18; i·(j×k) is a number.
Vector Algebra Class 12 Maths NCERT Solutions Miscellaneous Exercise: Related Exercises
The table lists the numbered 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 10.3 | Exercise 10.3 Solutions (Scalar or Dot Product) |
| Exercise 10.4 | Exercise 10.4 Solutions (Vector or Cross Product) |
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
NCERT Solutions Class 12 Maths: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
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 Misc with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 10 Vector Algebra Misc 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
Write down a unit vector in XY-plane, making an angle of 30∘ with the positive direction of x-axis.
Find the scalar components and magnitude of the vector joining the points P(x1, y1, z1) and Q(x2, y2, z2).
A girl walks 4 km towards west, then she walks 3 km in a direction 30∘ east of north and stops. Determine the girl's displacement from her initial point of departure.
If a = b + c, then is it true that |a| = |b| + |c|? Justify your answer.
Find the value of x for which x(i + j + k) is a unit vector.
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2i + 3j - k and b = i - 2j + k.
If a = i + j + k, b = 2i - j + 3k and c = i - 2j + k, find a unit vector parallel to the vector 2a - b + 3c.
Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a + b) and (a - 3b) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
The two adjacent sides of a parallelogram are 2i - 4j + 5k and i - 2j - 3k. Find the unit vector parallel to its diagonal. Also, find its area.
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ±(13, 13, 13).
Let a = i + 4j + 2k, b = 3i - 2j + 7k and c = 2i - j + 4k. Find a vector d which is perpendicular to both a and b, and c·d = 15.
The scalar product of the vector i + j + k with a unit vector along the sum of vectors 2i + 4j - 5k and λi + 2j + 3k is equal to one. Find the value of λ.
If a, b, c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b + c is equally inclined to a, b and c.
Prove that (a + b)·(a + b) = |a|2 + |b|2, if and only if a, b are perpendicular, given a ≠ 0, b ≠ 0.
If θ is the angle between two vectors a and b, then a·b ≥ 0 only when
(A) 0 < θ < π2
(B) 0 ≤ θ ≤ π2
(C) 0 < θ < π
(D) 0 ≤ θ ≤ π.
Let a and b be two unit vectors and θ is the angle between them. Then a + b is a unit vector if
(A) θ = π4
(B) θ = π3
(C) θ = π2
(D) θ = 2π3.
The value of i·(j×k) + j·(i×k) + k·(i×j) is
(A) 0 (B) -1 (C) 1 (D) 3.
If θ is the angle between any two vectors a and b, then |a·b| = |a×b| when θ is equal to
(A) 0 (B) π4 (C) π2 (D) π.
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 the Class 12 Maths Chapter 10 Miscellaneous Exercise?
Ans. The Miscellaneous Exercise of Class 12 Maths Chapter 10 Vector Algebra has 19 questions in the 2026-27 NCERT. Fifteen are short and long-answer problems that mix dot product, cross product and the section formula, and the last four (Q16 to Q19) are single-correct MCQs.
Ques. Is the Miscellaneous Exercise of Chapter 10 important for the CBSE board exam?
Ans. Yes. The Miscellaneous Exercise is the closest match to the this chapter's 5-mark long answer, because its questions chain several operations the way the board question does. Q9, Q10 and Q12 in particular reflect the section-formula and cross-product templates used in recent papers.
Ques. How do you find a vector perpendicular to two vectors with a given dot product in Q12?
Ans. Set d = k(a×b) , since the cross product is perpendicular to both a and b . Then use the condition c·d = 15 to solve for k. This gives d = 13(160i - 5j - 70k) .
Ques. Is | a | = | b | + | c | always true when a = b + c ?
Ans. No. In general |a| ≤ |b| + |c| by the triangle inequality, and equality holds only when b and c point in the same direction. This is the reasoning required in Q4 of the Miscellaneous Exercise.
Ques. How do I download the Class 12 Maths Chapter 10 Miscellaneous Exercise NCERT Solutions PDF?
Ans. Use the green download button on the NCERT Solutions Class 12 Maths card at the top of these notes to save the Collegedunia Class 12 Maths Chapter 10 Vector Algebra Miscellaneous Exercise NCERT Solutions PDF. The file is free, ad-free and mapped to the 2026-27 NCERT edition.
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