Vector Algebra Class 12 Notes for CBSE Boards and JEE

Vector algebra is one of the more scoring chapters in Class 12 Maths because most questions depend on a handful of formulas. These vector algebra class 12 notes bring those formulas and concepts together, based on NCERT Chapter 10, for CBSE board exams and JEE Main. Vectors also appear in Physics, in topics like work, torque and motion, so the chapter pays off beyond Maths. Read the key ideas below, then use the handwritten pages to practise. If you are preparing the whole syllabus, you can also read the Class 12 Maths notes for all chapters.

India's Best Exam Preparation for Class 12th - Download Now

Vector Algebra Class 12 Notes

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

Vector Algebra Class 12 Notes: Chapter Overview

A vector has both magnitude and direction, while a scalar has magnitude only. Distance and speed are scalars; displacement and velocity are vectors. In NCERT Class 12 Maths, the chapter covers:

• Scalars, vectors, magnitude and direction
• Direction cosines and direction ratios
• Types of vectors and the position vector
• Addition of vectors, multiplication by a scalar and the section formula
• Scalar (dot) product and projection of a vector
• Vector (cross) product and its applications

These ideas are the base for three-dimensional geometry, which follows this chapter.

Types of Vectors in Class 12 with Examples

Most questions start by identifying the type of vector given.

• Zero vector: magnitude 0 and no fixed direction, written $\vec{0}$.
• Unit vector: magnitude 1, found as $\hat{a} = \vec{a}/|\vec{a}|$. The vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ are unit vectors along the x, y and z axes.
• Equal vectors: same magnitude and same direction.
• Negative of a vector: same magnitude, opposite direction.
• Collinear (parallel) vectors: lie on the same line or parallel lines, so $\vec{a} = \lambda\vec{b}$ for some scalar λ.

Position Vector and Direction Cosines of a Vector

The position vector of P(x, y, z) is $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$ and its magnitude is $\sqrt{x^2+y^2+z^2}$. If a vector makes angles α, β and γ with the axes, its direction cosines are l = cos α, m = cos β and n = cos γ, and $l^2 + m^2 + n^2 = 1$. For $\vec{r} = a\hat{i}+b\hat{j}+c\hat{k}$, the direction cosines are $a/|\vec{r}|$, $b/|\vec{r}|$ and $c/|\vec{r}|$.

Vector Algebra Formulas for Class 12 and JEE

These are the vector algebra formulas that come up most often in Class 12 and JEE questions.

• Magnitude: $|\vec{a}| = \sqrt{a_1^2+a_2^2+a_3^2}$
• Triangle law: $\vec{AB} + \vec{BC} = \vec{AC}$
• Vector joining two points: $\vec{AB} = \vec{OB} - \vec{OA}$

Section Formula in Vector Algebra

The point dividing the line joining A($\vec{a}$) and B($\vec{b}$) in the ratio m : n internally has position vector $\dfrac{m\vec{b} + n\vec{a}}{m+n}$. For external division it is $\dfrac{m\vec{b} - n\vec{a}}{m-n}$, and the midpoint is $\dfrac{\vec{a}+\vec{b}}{2}$.

Dot Product and Cross Product

The dot product gives a scalar: $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1+a_2b_2+a_3b_3$. It is zero when the vectors are perpendicular, and the projection of $\vec{a}$ on $\vec{b}$ is $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$.

The cross product gives a vector perpendicular to both: $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$. It is zero when the vectors are parallel. The area of a parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ is $|\vec{a}\times\vec{b}|$, and the area of the triangle is half of that. Remember that $\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}$, while the dot product is commutative.

How to Find the Angle Between Two Vectors

Use $\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$. Find the dot product, find both magnitudes, divide, then take the inverse cosine. For example, if $\vec{a} = \hat{i}+2\hat{j}+2\hat{k}$ and $\vec{b} = 2\hat{i}-\hat{j}+2\hat{k}$, then $\vec{a}\cdot\vec{b} = 4$ and $|\vec{a}| = |\vec{b}| = 3$, so $\cos\theta = 4/9$.

Is Vector Algebra Important for JEE Main and CBSE Boards?

Yes. In the CBSE Class 12 paper, vectors and three-dimensional geometry are weighted together (14 marks in the current pattern, so check the latest CBSE blueprint). Most vector questions are formula-based, which makes the chapter easier to score. In JEE Main, vectors usually bring one or two direct questions, and the same ideas support 3D geometry. For JEE, also study the scalar triple product separately, as it goes beyond the CBSE syllabus. You can check the full list of topics in the JEE Main 2027 syllabus.

How to Revise Vector Algebra Class 12 Notes Faster

• Write all formulas on one page and revise it before every test.
• Remember the properties: the dot product is commutative, the cross product is not.
• Do not mix up sin θ (cross product) and cos θ (dot product).
• Solve the exercises of this chapter and compare your steps with the NCERT Solutions for Class 12 Maths.

eSaral is a Kota-based learning platform for JEE, NEET and Class 8–10 preparation, with the aim of making quality education reachable for every student in India. We hope these vector algebra class 12 notes help you remember the key points and revise with confidence. To read notes for every chapter, get the complete Class 12 Mathematics notes, and for video lectures and practice, download the eSaral app.