Vector Algebra Class 12 Notes - Class 12 Maths Ch 10

The Vector Algebra Class 12 Notes page compiles NCERT Class 12 Mathematics Chapter 10 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.

• CBSE Boards: 6 to 8 marks under the combined Vectors and Three Dimensional Geometry unit (14 marks total).
• JEE Main: 2 to 3 questions per shift, weightage close to 4 to 6 per cent of the Mathematics section.
• CUET-UG Maths: 1 to 2 direct questions on section formula, dot product and cross product.

Use the section sequence below to revise every formula, then move on to the matching NCERT Solutions for Chapter 10 on Collegedunia.

Every formula in this set has been verified against the 2026-27 NCERT print, the current CBSE marking schemes and the latest JEE Main session papers.

class 12 maths ncert notes chapter 10 Vector Algebra: Why This Chapter Pays Off

Vector Algebra is one of the highest scoring chapters in the Class 12 Mathematics syllabus because the formula set is small and the question patterns are predictable. The chapter rests on five operations: addition, scalar multiplication, the scalar (dot) product, the vector (cross) product and the scalar triple product.

A student who can write the dot and cross product formulas accurately and apply the section formula without sign errors will already secure the full 6 to 8 marks the Vector Algebra Class 12 Notes contributes to the board paper. Skipping Vector Algebra typically costs 10 to 14 marks once Chapter 11 Three Dimensional Geometry is added.

Vector Algebra Video Walkthrough

Source: Magnet Brains on YouTube

How will Collegedunia's NCERT Notes Help You Revise Vector Algebra?

The set is written for a final two-week revision window. Each section opens with the formula in the form an examiner expects, then a two-line explanation, then the standard exam application.

• NCERT-mapped explanations for every definition and solved example from the 2026-27 textbook.
• Boxed formula reminders for dot product, cross product, projection and the triple product.
• Physics-grounded intuition (force, velocity, work, torque) linked to Class 12 Physics.
• JEE-grade extensions: scalar triple product, vector area of a parallelogram, coplanarity condition.

Class 12 Maths Chapter 10 Vector Algebra: Topic-by-Topic Notes

The sub-topics below follow the NCERT order so the chapter can be revised straight through: position vectors, components along î, ĵ, k̂ , and the two products (dot and cross).

Scalars, Vectors and the Types You Must Recognise

A scalar is fully described by magnitude alone (mass, time, work). A vector needs both magnitude and direction (displacement, velocity, force, torque). Geometrically, a⃗ is a directed line segment of length |a⃗| . CBSE regularly tests the scalar-versus-vector distinction (speed versus velocity) as a 1-mark MCQ. The named types are:

• Zero vector 0⃗ : magnitude zero, direction indeterminate.
• Unit vector: â = a⃗|a⃗| , magnitude 1.
• Equal / negative vectors: same magnitude with same / opposite direction.
• Collinear vectors: parallel to the same line.
• Coplanar vectors: lie in one plane (used in the scalar triple product).
• Position vector: OP⃗ = r⃗ , from origin O to a point P.

Vector Addition and the Section Formula

By the triangle law, a⃗ + b⃗ = AC⃗ when a⃗ = AB⃗ and b⃗ = BC⃗ . The parallelogram law places two vectors at a common point as adjacent sides; the diagonal through that point is the sum. The magnitude of the resultant of two vectors inclined at angle θ is:

$$ |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta} $$

Vector addition is commutative and associative; scalar multiplication scales length as |λ a⃗| = |λ||a⃗| , with direction reversed when λ < 0 . If R divides AB (position vectors a⃗, b⃗ ) in the ratio m:n internally,

$$ \vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n} $$

For external division, replace each plus with a minus. The 1:1 case gives the midpoint r⃗ = 12(a⃗ + b⃗) . A 1 to 2 mark question on the section formula appears in nearly every recent CBSE Class 12 Maths paper.

Components Along i, j, k and Direction Cosines

Any vector in space decomposes uniquely as r⃗ = xî + yĵ + zk̂ , with |r⃗| = √x² + y² + z² . The cosines of the angles r⃗ makes with the positive x, y, z axes are the direction cosines l, m, n , and they satisfy:

$$ l^2 + m^2 + n^2 = 1 $$

Direction ratios are any triple proportional to l, m, n ; the components x, y, z themselves qualify. This is the algebraic bridge to Chapter 11 Three Dimensional Geometry.

Scalar (Dot) Product and Projection

The scalar product is

$$ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3 $$

It is commutative, distributive over addition, and equals zero exactly when the vectors are perpendicular. Two derived results cover Class 12 questions:

$$ \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}, \qquad \text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} $$

In Physics this is the formula for work done, W = F⃗ · d⃗ . For the foot of perpendicular, the angle between two vectors, or any projection, reach for the dot product first.

Vector (Cross) Product and Area

The vector product is a⃗ × b⃗ = |a⃗||b⃗|sinθ n̂ , with n̂ perpendicular to the plane of a⃗, b⃗ by the right-hand rule. In component form:

$$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} $$

The cross product is anti-commutative and equals zero exactly when the vectors are parallel. Parallelogram area with adjacent sides a⃗, b⃗ is |a⃗ × b⃗| ; triangle area with vertices A, B, C is 12|AB⃗ × AC⃗| . Physically, the cross product underpins torque τ⃗ = r⃗ × F⃗ and angular momentum L⃗ = r⃗ × p⃗ .

Scalar Triple Product and Coplanarity (JEE Extension)

Outside the current 2026-27 NCERT exercise but central to JEE Main case-studies:

$$ [\vec{a} \; \vec{b} \; \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} $$

The value equals the volume of the parallelepiped on edges a⃗, b⃗, c⃗ ; the three vectors are coplanar exactly when [a⃗ b⃗ c⃗] = 0 . When a problem names four points and asks for coplanarity, build three vectors from one point to the other three and test the triple product.

Vector Algebra Class 12: Most Repeated Board Questions

The four solved problems below cover roughly 70 per cent of the question flavours CBSE has set on the Vector Algebra Class 12 Notes over the last five years.

Ques. Find the unit vector in the direction of a⃗ = 2î + 3ĵ - k̂ . (2019, 2022, 2024)

[1-Mark] Ans. |a⃗| = √4+9+1 = √14 , so â = 1√14(2î + 3ĵ - k̂) .

Ques. Find the position vector of R which divides the line joining A(2î - ĵ + k̂) and B(î + 2ĵ - 3k̂) in the ratio 2:1 internally. (2018, 2023)

[2-Mark] Ans. By the section formula, r⃗ = 2b⃗ + a⃗3 = 2(î + 2ĵ - 3k̂) + (2î - ĵ + k̂)3 = 4î + 3ĵ - 5k̂3 .

Ques. Find the angle between a⃗ = î + ĵ - k̂ and b⃗ = î - ĵ + k̂ . (2020, 2024)

[3-Mark] Ans. a⃗·b⃗ = 1 - 1 - 1 = -1 and |a⃗| = |b⃗| = √3 , so cosθ = -13 and θ = cos^-1(-13) .

Ques. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), C(1, 5, 5). (2017, 2022, 2024)

[3-Mark / 5-Mark] Ans. AB⃗ = î + 2ĵ + 3k̂ and AC⃗ = 4ĵ + 3k̂ . Then AB⃗×AC⃗ = -6î - 3ĵ + 4k̂ , with magnitude √61 . Hence Area = 12√61 square units.

Vector Algebra in Recent CBSE Class 12 Papers: A 5-Year Analysis

Across CBSE Class 12 Maths papers from 2019 to 2024, one MCQ on direction cosines or types of vectors and one 2-mark item on the section formula or a projection appear almost every year, alongside a 3-mark numerical on the angle between two vectors or the area of a triangle. The 5-mark slot is shared with Three Dimensional Geometry.

Vector Algebra Beyond Boards: JEE Main and CUET Weightage

Vectors is one of the few Class 12 chapters where the boards, JEE Main and CUET-UG draw on the same machinery at different difficulty levels. The exam-comparison table below leads with the latest held edition.

Vector Algebra Class 12: Common Mistakes to Avoid

Most marks lost in the Vector Algebra Class 12 Notes are not from gaps in knowledge but from a small number of recurring slips. Addressing the four below lifts accuracy by several marks per paper.

• Confusing dot product with cross product: dot returns a scalar and uses cosθ ; cross returns a vector and uses sinθ . Mixing them is the most common 2-mark loss.
• Sign error in the cross product: the ĵ component of a⃗×b⃗ carries a minus in the determinant expansion. Forgetting it flips the answer.
• Wrong projection formula: the projection of a⃗ on b⃗ is a⃗·b⃗|b⃗| , with one magnitude in the denominator (not two).
• Forgetting the 12 in triangle area: cross product magnitude gives parallelogram area; halve it for a triangle.

Related Resources for Class 12 Maths Chapter 10 Vector Algebra

• NCERT Solutions: Chapter 10 Vector Algebra
• Formula Sheet: Chapter 10 Vector Algebra
• NCERT Book PDF: Chapter 10 Vector Algebra
• Handwritten Notes: Chapter 10 Vector Algebra
• Exemplar Solutions: Chapter 10 Vector Algebra

NCERT Notes for Class 12 Maths: All Chapters

Also Check: chapter-wise revision sets for the rest of the Class 12 Mathematics syllabus.

Vector Algebra Class 12 Notes: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.

Vector Algebra Class 12 Notes - Quick Summary

• The Vector Algebra Class 12 Notes cover every section of Class 12 Mathematics Chapter 10 Vector Algebra, aligned to the 2026-27 NCERT print.
• The Vector Algebra Class 12 Notes include formal definitions, solved examples and end-of-section formula recap suitable for board and JEE Main preparation.
• The Vector Algebra Class 12 Notes are downloadable as a free PDF and follow the notation of the official NCERT textbook line for line.

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.

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:

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:

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 these 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:

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

The Vector Algebra Class 12 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:

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 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:

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:

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.