Class 12 · Chapter 10 · Vectors and Three-Dimensional Geometry unit (14 of 80 marks)

Vector Algebra Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 49 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

49 questions
15 multiple choice, 4 assertion–reason, 13 very short answer, 10 short answer, 4 long answer, 3 case study
About 7 hours to master

Unit IV Vectors and Three-Dimensional Geometry: 14 marks of the 80-mark paper (shared with Chapter 11) — CBSE Curriculum 2025-26 Mathematics (041) (official PDF).

NCERT textbook: Chapter 10 on ncert.nic.in (PDF)

Revision notes

Vector Algebra — revision notes

1. Basics

$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ : magnitude $| \vec{r} | = \sqrt{x^{2} + y^{2} + z^{2}}$ , unit vector $\hat{r} = \frac{\vec{r}}{| \vec{r} |}$ .
$\vec{A B} = \vec{b} - \vec{a}$ (position vector of head minus tail).
Direction cosines $l = \frac{x}{r}, m = \frac{y}{r}, n = \frac{z}{r}$ with $l^{2} + m^{2} + n^{2} = 1$ ; direction ratios are any multiple $(x, y, z)$ .
Collinear (parallel) vectors: $\vec{b} = λ \vec{a}$ , i.e. components in the same ratio.

2. Section formula

Point dividing $A B$ internally in $m : n$ : $\frac{n \vec{a} + m \vec{b}}{m + n}$ ; externally: $\frac{m \vec{b} - n \vec{a}}{m - n}$ ; midpoint $\frac{\vec{a} + \vec{b}}{2}$ .

3. Scalar (dot) product

$\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ ; $θ \in [0, π]$ .
$\vec{a} ⊥ \vec{b} ⟺ \vec{a} \cdot \vec{b} = 0$ ; $\vec{a} \cdot \vec{a} = | \vec{a} |^{2}$ .
Projection of $\vec{a}$ on $\vec{b}$ : $\frac{\vec{a} \cdot \vec{b}}{| \vec{b} |}$ ; projection vector $(\frac{\vec{a} \cdot \vec{b}}{| \vec{b} |^{2}}) \vec{b}$ .
$| \vec{a} \pm \vec{b} |^{2} = | \vec{a} |^{2} + | \vec{b} |^{2} \pm 2 \vec{a} \cdot \vec{b}$ — the key tool for magnitude problems.

4. Vector (cross) product

$\vec{a} \times \vec{b} = | \vec{a} | | \vec{b} | \sin θ \hat{n}$ (right-hand rule) $= | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} |$ .
$\vec{b} \times \vec{a} = - \vec{a} \times \vec{b}$ ; $\vec{a} \times \vec{a} = \vec{0}$ ; $\vec{a} ∥ \vec{b} ⟺ \vec{a} \times \vec{b} = \vec{0}$ .
Area of parallelogram $= | \vec{a} \times \vec{b} |$ (adjacent sides) $= \frac{1}{2} | {\vec{d}}_{1} \times {\vec{d}}_{2} |$ (diagonals); area of triangle $A B C = \frac{1}{2} | \vec{A B} \times \vec{A C} |$ .
Unit vector perpendicular to $\vec{a}$ and $\vec{b}$ : $\pm \frac{\vec{a} \times \vec{b}}{| \vec{a} \times \vec{b} |}$ . Also $| \vec{a} \times \vec{b} |^{2} + (\vec{a} \cdot \vec{b})^{2} = | \vec{a} |^{2} | \vec{b} |^{2}$ .

Worked example 1

$| \vec{a} | = 2$ , $| \vec{b} | = 3$ , $\vec{a} \cdot \vec{b} = 3$ : $| \vec{a} - \vec{b} |^{2} = 4 + 9 - 6 = 7$ , so $| \vec{a} - \vec{b} | = \sqrt{7}$ ; $\cos θ = \frac{1}{2}$ , $θ = \frac{π}{3}$ .

Worked example 2

Area of the triangle with vertices $(0, 0, 0)$ , $(1, 2, 0)$ , $(0, 1, 3)$ : $(\hat{i} + 2 \hat{j}) \times (\hat{j} + 3 \hat{k}) = 6 \hat{i} - 3 \hat{j} + \hat{k}$ , area $= \frac{1}{2} \sqrt{46}$ .

Worked example 3

Point dividing $A (2, 1, 3)$ , $B (5, 7, 0)$ in $1 : 2$ : $\frac{2 (2, 1, 3) + (5, 7, 0)}{3} = (3, 3, 2)$ .

Common errors

Middle term of the cross-product determinant: it is $- \hat{j} (a_{1} b_{3} - a_{3} b_{1})$ .
Using $\frac{m \vec{a} + n \vec{b}}{m + n}$ (weights swapped) in the section formula.
Giving only one sign for a unit perpendicular vector, or the wrong one when a condition (acute/obtuse with an axis) is given.
Confusing the scalar projection (a number) with the projection vector.
Writing $| \vec{a} + \vec{b} | = | \vec{a} | + | \vec{b} |$ .

Board-exam tips

For “find the angle” questions, write $\cos θ = \dots$ and the final $θ = \cos^{- 1} (\cdot)$ if it is not standard.
Scalar triple product and coplanarity are not in the current syllabus.

Topics in this chapter: Vectors: magnitude, direction and types · Direction cosines and direction ratios · Addition, scalar multiple and section formula · Scalar (dot) product and projection · Vector (cross) product and area.

Formulas for this chapter: Chapter 10 on the Class 12 formula sheet (free PDF and A3 poster).

Vector Algebra Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 215 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceScalar (dot) product

Vectors $\vec{a}$ and $\vec{b}$ have $| \vec{a} | = 2$ , $| \vec{b} | = 5$ and $\vec{a} \cdot \vec{b} = 5 \sqrt{3}$ . What is the angle between them?

Show worked answer

$\begin{aligned} \cos θ = \frac{5 \sqrt{3}}{2 \times 5} = \frac{\sqrt{3}}{2} \\ \Rightarrow & θ = \frac{π}{6} \end{aligned}$ A1
Correct option: (a) $\frac{π}{6}$

Q2·1 mark·Multiple choiceVectors, magnitude and direction

The unit vector in the direction of $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ is

Q3·1 mark·Assertion–reasonVector (cross) product

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): $\hat{i} \times \hat{j} = \hat{k}$ .

Reason (R): $\hat{i} \cdot \hat{j} = 0$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerScalar (dot) product

Find $λ$ if $\vec{a} = 2 \hat{i} + λ \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ are perpendicular.

Show worked answer

Answer:

λ = \frac{5}{2}

$\begin{aligned} \vec{a} \cdot \vec{b} & = 2 - 2 λ + 3 \\ = 0 \end{aligned}$ M1
$λ = \frac{5}{2}$ A1

Q5·2 marks·Short answerVector (cross) product

Find the area of the parallelogram whose adjacent sides are $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $\vec{b} = 3 \hat{i} - 2 \hat{j} + \hat{k}$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerScalar (dot) product

Find the projection of $\vec{a} = 3 \hat{i} - \hat{j} + 4 \hat{k}$ on $\vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ .

Show worked answer

Answer:

3

$\vec{a} \cdot \vec{b} = 3 - 2 + 8 = 9$ M1
$| \vec{b} | = \sqrt{1 + 4 + 4} = 3$ A1
Projection $= \frac{\vec{a} \cdot \vec{b}}{| \vec{b} |} = 3$ A1

Q7·3 marks·Short answerVectors, magnitude and direction

Find a vector of magnitude $9$ in the direction of $2 \hat{i} - \hat{j} + 2 \hat{k}$ , and the direction cosines of that direction.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerVector (cross) product

$A (1, 0, 2)$ , $B (3, 1, 4)$ and $C (2, 3, 1)$ are the vertices of a triangle. Find (a) $\vec{A B}$ and $\vec{A C}$ , (b) the area of $△ A B C$ , (c) $\cos A$ .

Show worked answer

Answer: Area

\frac{3 \sqrt{10}}{2}

;

\cos A = \frac{1}{\sqrt{11}}

(a) $\vec{A B} = 2 \hat{i} + \hat{j} + 2 \hat{k}$ , $\vec{A C} = \hat{i} + 3 \hat{j} - \hat{k}$ A1
(b) $\vec{A B} \times \vec{A C} = - 7 \hat{i} + 4 \hat{j} + 5 \hat{k}$ M1
$\begin{aligned} | \vec{A B} \times \vec{A C} | & = \sqrt{49 + 16 + 25} \\ = \sqrt{90} \\ = 3 \sqrt{10} \end{aligned}$ A1
Area $= \frac{1}{2} \times 3 \sqrt{10} = \frac{3 \sqrt{10}}{2}$ square units A1
(c) $\begin{aligned} \cos A & = \frac{\vec{A B} \cdot \vec{A C}}{| \vec{A B} | | \vec{A C} |} \\ = \frac{2 + 3 - 2}{3 \sqrt{11}} \\ = \frac{1}{\sqrt{11}} \end{aligned}$ A1

Q9·5 marks·Long answerVector (cross) product

For $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}$ : (a) find $\vec{a} \cdot \vec{b}$ and the cosine of the angle between them; (b) find $\vec{a} \times \vec{b}$ and a unit vector perpendicular to both; (c) check that $\vec{a} \times \vec{b}$ is perpendicular to $\vec{a}$ .

Section E style · Case study (4 marks)

Q10·4 marks·Case studyVectors, magnitude and direction

Drone flight. A drone flies in a straight line from $P (1, 2, 0)$ to $Q (4, 6, 0)$ , then straight up to $R (4, 6, 5)$ (units: metres).

Write $\vec{P Q}$ and find its length. [1 mark]
Find the straight-line distance from $P$ to $R$ . [1 mark]
Find the angle between $\vec{P Q}$ and $\vec{P R}$ . [2 marks]

Show worked answer

$\vec{P Q} = 3 \hat{i} + 4 \hat{j}$ , length $5$ m A1
$\vec{P R} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k}$ , $\begin{aligned} | \vec{P R} | & = \sqrt{50} \\ = 5 \sqrt{2} \end{aligned}$ m A1
$\begin{aligned} \cos θ & = \frac{9 + 16 + 0}{5 \times 5 \sqrt{2}} \\ = \frac{1}{\sqrt{2}} \end{aligned}$ M1
$θ = \frac{π}{4}$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

Find magnitudes, unit vectors, direction cosines, section-formula points and simple dot/cross products.

Read first: 1. Basics; 2. Section formula 6 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

Use dot and cross products for angles, projections, perpendicularity, collinearity and areas at 2- and 3-mark level.

Read first: 3. Scalar (dot) product; 4. Vector (cross) product 23 practice questions · checkpoint: 4 questions, 8 marks, pass 75%

Practise step 2

Step 3

Full marks on long answers

Write complete multi-step vector solutions (triangle and parallelogram problems, case studies with forces or displacements).

Read first: Worked examples 1-3; 4. Vector (cross) product (areas) 4 practice questions · checkpoint: 3 questions, 14 marks, pass 70%

Practise step 3

Step 4

95+ stretch (HOTS)

Work with vectors given only by magnitudes and angles, minimisation and perpendicular-component arguments.

Read first: 3. Scalar (dot) product (|a ± b|² identity) 6 practice questions · checkpoint: 3 questions, 13 marks, pass 60%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 49 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceVectors: magnitude, direction and types

A vector of magnitude $14$ in the direction opposite to $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ is

(a) $4 \hat{i} - 6 \hat{j} + 12 \hat{k}$
(b) $- 4 \hat{i} + 6 \hat{j} - 12 \hat{k}$
(c) $- 28 \hat{i} + 42 \hat{j} - 84 \hat{k}$
(d) $- 2 \hat{i} + 3 \hat{j} - 6 \hat{k}$

Q2·1 mark·Multiple choiceAddition, scalar multiple and section formula

The position vectors of $A$ and $B$ are $\hat{i} + 2 \hat{j} - \hat{k}$ and $6 \hat{i} - 3 \hat{j} + 4 \hat{k}$ . The position vector of the point dividing $A B$ internally in the ratio $2 : 3$ is

(a) $\frac{7}{2} \hat{i} - \frac{1}{2} \hat{j} + \frac{3}{2} \hat{k}$
(b) $4 \hat{i} - \hat{j} + 2 \hat{k}$
(c) $3 \hat{i} + \hat{k}$
(d) $3 \hat{i} + 2 \hat{k}$

Q3·1 mark·Multiple choiceVector (cross) product and area

$(2 \hat{i} - \hat{j}) \times (\hat{i} + 3 \hat{k})$ equals

(a) $- 3 \hat{i} + 6 \hat{j} + \hat{k}$
(b) $3 \hat{i} + 6 \hat{j} - \hat{k}$
(c) $- 3 \hat{i} - 6 \hat{j} + \hat{k}$
(d) $2 \hat{i} - 3 \hat{k}$

Worked solution·2 marks·Very short answerDirection cosines and direction ratios

Find the direction cosines of the vector joining $A (1, 2, - 3)$ to $B (- 1, - 2, 1)$ .

Full worked solution Free, no sign-in

$\vec{A B} = - 2 \hat{i} - 4 \hat{j} + 4 \hat{k}$ , $| \vec{A B} | = 6$ A1
Direction cosines $- \frac{1}{3}, - \frac{2}{3}, \frac{2}{3}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S71 · Problem S78 · Problem S21 · Problem S77. All 22 →

Knowledge organiser: Vector Algebra on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Magnitude of a vector. All formulas A–Z

Where marks are lost in Vector Algebra

Sign error in the ĵ-term of a cross product. Fix: write −ĵ(a₁b₃ − a₃b₁) explicitly, then check that the answer is perpendicular to both vectors (dot = 0).
Section formula with weights swapped. Fix: the point dividing AB in m : n is (n·a + m·b)/(m + n) — the weight of b is the part nearer A.
Only one of ± for a unit perpendicular vector. Fix: give ±, then apply any stated condition (e.g. acute angle with the z-axis) to choose.
Area of triangle given as |AB × AC| (forgetting ½), or parallelogram from diagonals without ½. Fix: write the formula before substituting.
Treating magnitudes as additive: |a + b| = |a| + |b|. Fix: square and use |a + b|² = |a|² + |b|² + 2a·b.
Stopping at cos θ = 4/9 without stating θ. Fix: finish with θ = cos⁻¹(4/9).

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Examiner Insights: common mistakes in CBSE Class 12 Maths, with fixes (our analysis of public sources) →

Vector Algebra in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 13, 15, 24
Sample paper 2 (Board standard): questions 16, 19, 34
Sample paper 3 (Above board): questions 13, 14, 24
Sample paper 4 (Above board): questions 16, 19, 25
Sample paper 5 (95+ challenge): questions 13, 19, 24

Want it against the clock? Take a timed 30-mark chapter test on Vector Algebra, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on vectors and three-dimensional geometry.

Vector Algebra: common questions

What is in CBSE Class 12 Maths Chapter 10, Vector Algebra?

The topics are: Vectors: magnitude, direction and types, Direction cosines and direction ratios, Addition, scalar multiple and section formula, Scalar (dot) product and projection, Vector (cross) product and area. This page has revision notes for each, and 49 board-style practice questions with step-marked solutions.

How many marks is Vector Algebra worth?

It is in the Vectors and Three-Dimensional Geometry unit, which carries 14 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Vector Algebra?

Sign error in the ĵ-term of a cross product. Fix: write −ĵ(a₁b₃ − a₃b₁) explicitly, then check that the answer is perpendicular to both vectors (dot = 0). Section formula with weights swapped. Fix: the point dividing AB in m : n is (n·a + m·b)/(m + n) — the weight of b is the part nearer A.

Are the Vector Algebra important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Vector Algebra — revision notes

1. Basics

2. Section formula

3. Scalar (dot) product

4. Vector (cross) product

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Vector Algebra Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Vector Algebra

Vector Algebra in our sample papers

Vector Algebra: common questions

Work through the Route to 95 with your progress saved