Class 12 Maths NCERT Solutions Chapter 10 Vector Algebra | Boundless Maths
NCERT Solutions Class 12 Maths Chapter 10 · Unit IV · Vectors & 3D Geometry

Class 12 Maths NCERT Solutions Chapter 10: Vector Algebra

Free, step-by-step Class 12 Maths NCERT Solutions Chapter 10 for all four exercises — basic concepts and types of vectors, addition of vectors and the section formula, the scalar (dot) product, and the vector (cross) product — plus the Miscellaneous Exercise. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

5Exercises (incl. Misc.)
73Total Questions
2026-27CBSE Syllabus
100%Solved

Class 12 Maths NCERT Solutions Chapter 10 — Overview

Vector Algebra introduces quantities that carry both magnitude and direction — displacement, velocity, force — and gives you a precise algebra for working with them, using the unit vectors î, ĵ, k̂ to write any vector in component form. It's the shortest chapter in the Vectors & 3D Geometry unit by concept count, but everything in it is load-bearing: Chapter 11 (Three Dimensional Geometry) leans on direction cosines, the dot product and the cross product constantly, just applied to lines and planes instead of free vectors.

The chapter builds in a clean, deliberate sequence across its four exercises: basic definitions and types of vectors first, then addition and the section formula for dividing a line segment in a given ratio, then the scalar (dot) product for angles and projections, and finally the vector (cross) product for areas and perpendicularity. The Miscellaneous Exercise mixes all four together — exactly the way CBSE actually tests this chapter in a real paper.

How the Chapter Builds

One Idea Leads to the Next

1

Basic Concepts & Types

Magnitude, direction, unit/zero/collinear/equal vectors — the vocabulary everything else depends on. Exercise 10.1.

→
2

Addition & Section Formula

Component form, vector addition, unit vectors, and dividing a segment in a given ratio. Exercise 10.2.

→
3

Scalar (Dot) Product

a·b gives a scalar — used for the angle between vectors, perpendicularity, and projection. Exercise 10.3.

→
4

Vector (Cross) Product

a×b gives a vector perpendicular to both — used for areas of triangles/parallelograms and finding normal vectors. Exercise 10.4 & Misc.

Quick Reference

Important Formulas — Chapter 10

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below have the full printable version for the whole syllabus.

Basic Concepts (§10.2, §10.5)

Magnitude of a vector

|\vec{r}|=\sqrt{x^2+y^2+z^2}

For \vec{r}=x\hat{i}+y\hat{j}+z\hat{k}.

Unit vector in the direction of a

\hat{a}=\dfrac{1}{|\vec{a}|}\vec{a}

Always has magnitude 1, same direction as a.

Direction cosines

l=\dfrac{a_1}{|\vec{r}|},\ m=\dfrac{a_2}{|\vec{r}|},\ n=\dfrac{a_3}{|\vec{r}|}

Satisfy l^2+m^2+n^2=1 — carries straight into Chapter 11.

Addition & Section Formula (§10.4, §10.5.3)

Vector joining two points

\vec{P_1P_2}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}

The building block for almost every question in Exercise 10.2.

Section formula (internal)

\vec{r}=\dfrac{m\vec{b}+n\vec{a}}{m+n}

Point dividing PQ internally in the ratio m : n.

Section formula (external)

\vec{r}=\dfrac{m\vec{b}-n\vec{a}}{m-n}

Same ratio, but dividing the segment externally.

Scalar (Dot) Product (§10.6.1)

Definition

\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta

A scalar result — never a vector.

Angle between two vectors

\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}

Perpendicular exactly when \vec{a}\cdot\vec{b}=0.

Projection of a on b

\dfrac{1}{|\vec{b}|}(\vec{a}\cdot\vec{b})

A scalar length — the "shadow" of a along b's direction.

Vector (Cross) Product (§10.6.3)

Definition

\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}

A vector, perpendicular to both a and b.

Area of a triangle

\text{Area}=\dfrac12|\vec{AB}\times\vec{AC}|

Half the magnitude of the cross product of two sides.

Area of a parallelogram

\text{Area}=|\vec{a}\times\vec{b}|

The full magnitude, using the two adjacent side vectors.

Decision Guide

Dot Product or Cross Product — Which One?

The single most common point of confusion in this chapter, resolved by asking what kind of answer the question actually wants.

What the question asks forUse thisWhy
The angle between two vectors, or to check perpendicularityDot producta·b = 0 exactly when the vectors are perpendicular — a direct algebraic test.
The projection or "shadow length" of one vector on anotherDot productProjection is defined using the dot product divided by the target vector's magnitude.
A vector perpendicular to two given vectorsCross productOnly the cross product returns a vector at all — the dot product always returns a scalar.
The area of a triangle or parallelogramCross product|a×b| directly equals the parallelogram's area, and half of it gives the triangle's area.
Checking whether two vectors are parallel/collinearCross producta×b = 0 exactly when the vectors are parallel (or one is the zero vector).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all four exercises.

  • Confusing collinear vectors with equal vectors. Collinear only requires parallel direction lines (magnitude and sense can differ); equal vectors need the same magnitude and the same direction.
  • Forgetting the absolute value in the angle-between-vectors formula for the cross product's sine version, and in the direction-ratio angle formula that carries into Chapter 11.
  • Sign errors when expanding the 3×3 determinant for a cross product — a single misplaced sign flips the whole answer's direction.
  • Treating the dot product and cross product as interchangeable — reaching for a scalar formula when the question genuinely wants a vector (or vice versa).
  • Mixing up internal and external section formula — using a plus sign in the denominator when the ratio requires external division (which needs a minus).
  • Forgetting to normalise when a question specifically asks for a unit vector, and leaving the answer as a non-unit direction vector instead.
  • Not checking whether the vectors given are actually nonzero before applying a perpendicularity or parallelism test — both a·b = 0 and a×b = 0 also hold trivially if either vector is the zero vector.
  • Losing track of which vector is a and which is b when the formula isn't symmetric — the cross product in particular is anti-commutative: a×b = −(b×a).
Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 10 — Choose an Exercise

10.1

Exercise 10.1

Scalars vs vectors, and coinitial, equal, and collinear vectors identified from a figure · 5 questions

Solve Exercise 10.1 →
10.2

Exercise 10.2

Vector addition, magnitude, unit vectors, direction cosines, collinearity and the section formula · 19 questions

Solve Exercise 10.2 →
10.3

Exercise 10.3

The scalar (dot) product — angle between vectors, projection, and perpendicularity · 18 questions

Solve Exercise 10.3 →
10.4

Exercise 10.4

The vector (cross) product — perpendicular unit vectors, and area of a triangle and a parallelogram · 12 questions

Solve Exercise 10.4 →
M

Miscellaneous Exercise

Mixed questions combining addition, the section formula, the dot product and the cross product · 19 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

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Common Questions

FAQs — Class 12 Maths NCERT Solutions Chapter 10

Quick answers about Chapter 10, Vector Algebra.

How many exercises are there in Chapter 10, Vector Algebra?

There are four main exercises — 10.1 to 10.4 — plus a Miscellaneous Exercise, totalling 73 questions across basic vector concepts, addition and the section formula, the scalar (dot) product, and the vector (cross) product.

Is this chapter important for the board exam?

Yes — it's tested directly, and it's the essential foundation for Chapter 11 (Three Dimensional Geometry), where direction cosines, the dot product and the cross product all reappear applied to lines in space.

What is the difference between the scalar (dot) product and the vector (cross) product?

The dot product of two vectors is a scalar (a·b = |a||b|cosθ), and is used to find the angle between vectors, test perpendicularity, and compute projections. The cross product is a vector (a×b = |a||b|sinθ n̂), perpendicular to both original vectors, and is used to find a vector perpendicular to a plane and to compute areas of triangles and parallelograms.

What should I revise before starting this chapter?

Vector Algebra is largely self-contained and a good entry point into the Vectors & 3D Geometry unit, but comfort with basic coordinate geometry and trigonometric ratios (for angle calculations) will make the dot and cross product sections much smoother.

Where can I find the official NCERT textbook for this chapter?

Chapter 10, Vector Algebra, is from the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this site follow the questions exactly as they appear there.
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