NCERT Class 12 Maths Chapter 10 Vector Algebra PDF – Concepts, Formulas, and Download Link

Chapter 10 of Class 12 Maths, Vector Algebra, is all about understanding vectors, their operations, and how they apply to real-world problems in geometry and physics. This chapter introduces the basic concepts like magnitude, direction, addition, scalar and vector products, and how they are represented algebraically. NCERT explains each topic with proper examples, making it

Chapter 10 of Class 12 Maths, Vector Algebra, is all about understanding vectors, their operations, and how they apply to real-world problems in geometry and physics. This chapter introduces the basic concepts like magnitude, direction, addition, scalar and vector products, and how they are represented algebraically. NCERT explains each topic with proper examples, making it easier to understand and apply in both school exams and entrance tests like JEE.

I decided to write about Vector Algebra because many students struggle with visualising vectors. When I was in Class 12, I also found it difficult in the beginning, especially when it came to 3D vectors and cross products. But once I understood the concept of direction and magnitude clearly, the whole chapter started making sense. It’s an important chapter not only for board exams but also for future topics in engineering, physics, and computer science. A well-organised PDF with clear explanations helps students revise better, especially during last-minute preparations. That’s why I’ve included a download option too.

Important Concepts in Chapter 10: Vector Algebra

Vector Algebra forms the foundation for topics in both pure and applied mathematics. It helps in representing and solving problems involving force, motion, and geometry. Let’s look at the key topics covered in this chapter:

1. Basics of Vectors

A vector has both magnitude and direction
Represented usually by an arrow or as a⃗\vec{a}a in mathematics
Vectors can be position vectors or free vectors

2. Types of Vectors

Zero vector: Magnitude is zero
Unit vector: Magnitude is 1
Collinear vectors: Lie on the same line or are parallel
Equal vectors: Same magnitude and direction

3. Vector Operations

Addition of vectors: Triangle law and parallelogram law
Subtraction of vectors: Using addition of negative vectors
Multiplication by a scalar: Changes magnitude, not direction (unless scalar is negative)

4. Components of Vector

5. Scalar (Dot) Product

Used to find angle between two vectors or to check perpendicularity

6. Vector (Cross) Product

Result is a vector perpendicular to both a and b
Important in physics (e.g., torque)

Why You Should Focus on Chapter 10 for Boards and JEE

Vector Algebra questions are direct and usually carry 4 to 6 marks in CBSE board exams. Questions like proving two vectors are perpendicular, finding direction ratios, or applying scalar product are very common. Also, JEE Main and Advanced always include at least one question from this topic. I remember solving a question in JEE that was directly from NCERT Example 10.5.

Students often ignore diagrams while solving vector problems, but they are actually very useful for understanding direction and checking your answer. So draw diagrams wherever possible. Also, keep revising formulas because remembering them makes problem-solving faster during exams.

Download NCERT Class 12 Maths Chapter 10 Vector Algebra PDF

You can easily download the full chapter PDF for offline study.

NCERT Class 12 English Vistas Chapter 6: Memories of Childhood PDF Download & Complete Summary

Memories of Childhood is Chapter 6 in the Class 12 English Vistas textbook by NCERT. What makes this chapter special is that it’s not a single story—it’s made up of two autobiographical excerpts written by two women, Zitkala-Sa and Bama. Both narrate incidents from their early lives that reflect the discrimination they faced because of

Memories of Childhood is Chapter 6 in the Class 12 English Vistas textbook by NCERT. What makes this chapter special is that it’s not a single story—it’s made up of two autobiographical excerpts written by two women, Zitkala-Sa and Bama. Both narrate incidents from their early lives that reflect the discrimination they faced because of their identity. While Zitkala-Sa was a Native American who suffered cultural oppression, Bama grew up as a Dalit in South India, dealing with caste-based discrimination. These two stories are placed together in the chapter to show that such issues exist in many parts of the world.

I decided to write about this chapter because it’s different from the others in the book. It’s not fiction, but a real account of how young children feel when they are treated unfairly just because of their background. When I first read Bama’s story, I couldn’t stop thinking about how common such behaviour still is in many Indian towns. These stories make us more aware and sensitive. As students, we need to understand not just grammar and vocabulary but also social realities. This chapter is an important step towards building that understanding.

What is “Memories of Childhood” About?

This chapter includes two short autobiographical narratives that focus on childhood memories shaped by discrimination and injustice.

Part One: The Cutting of My Long Hair by Zitkala-Sa

This section is written by a Native American woman who was taken to a missionary school where they tried to make her behave like a “white” child. She describes how humiliating it was when her long hair was forcibly cut, which in her culture was a sign of defeat. She feels powerless, angry, and alien in a place that tries to erase her identity.

Part Two: We Too Are Human Beings by Bama

Bama’s story takes place in a village in Tamil Nadu. She is a young Dalit girl who notices how upper-caste people treat her community with disrespect. She sees a man carrying food for his master without touching it, just because of his caste. When she narrates this incident to her brother, he tells her about caste discrimination and advises her to study hard to earn respect and equality.

Main Characters:

Zitkala-Sa – A Native American girl forced to give up her culture at a white-run school
Bama – A young Dalit girl who slowly becomes aware of social injustice

Key Themes:

Discrimination based on race and caste
Childhood trauma and awareness
Loss of identity
Education as a tool for change

Why This Chapter Matters for Students

Real stories: It helps students connect emotionally to real-life problems faced by children
Relevant to society: Topics like caste and cultural identity are still important in today’s India
CBSE exam favourite: Frequently asked in long and value-based questions
Builds empathy: Helps students look beyond textbooks and understand people around them better
Teaches resilience: Both writers turned their pain into power through education and awareness