CBSE Class 12 · Mathematics
Vector Algebra
Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.
This lesson follows the official chapter “Vector Algebra” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 12
- Medium level
- 8 concepts
1Some Basic Concepts
Basic concepts: a vector has magnitude and direction as taught, written as a directed segment or as components. a = 2i+3j has magnitude sqrt(13). A vector is not a lone number (that is a scalar). Position vector is from the origin as taught.
Calling 5 a vector because it is “big” is a miss.
Figure. A vector is an arrow: size and direction from a chosen origin. OP is the position vector of P.
How it works
- Name magnitude and direction, or the componentsThe vector.
- Keep a scalar as a number without directionThe split.
- Write |a| as the lengthThe size.
2Types of Vectors
Types: zero, unit, equal, collinear, negative as taught. A unit vector has length 1; a-hat = a/|a|. Type is a look-test. The zero vector has no unique direction as framed.
A long arrow called unit because it “points well” is a miss.
Figure. Equal vectors match length and direction. A unit vector has length 1. The zero vector has no direction to draw as an arrow.
How it works
- Name zero / unit / equal / collinearThe type.
- Run the length or parallel testIn or out.
- Keep a-hat as a/|a|The unit.
3Addition of Vectors
Addition: triangle or parallelogram as taught; components add matching seats. (2i+3j)+(4i-j)=6i+2j. Addition is tip-to-tail or component-sum, not multiplying lengths.
Adding |a|+|b| as |a+b| is a miss unless they line up.
Figure. Tip-to-tail, or the diagonal of the completed parallelogram. a + b ends at the far corner.
How it works
- Add tip-to-tail or add componentsThe sum.
- Keep |a+b| ≤ |a|+|b| as the school cautionA triangle.
- Refuse |a|+|b| as the default sumWrong size.
Add components
(2i+3j)+(4i-j).
- i-parts2+4=6
- j-parts3+(-1)=2
- Sum6i+2j
Pro tip. Matching seats.
4Multiplication of a Vector by a Scalar
Scalar multiplication: k a stretches (or flips if k<0) as taught. 3(2i+3j)=6i+9j. Scalar multiply is a stretch, not a second vector-add.
Adding 3 to each component of 2i+3j as “3 times” is a miss.
Figure. A positive scalar stretches on the same ray. A negative scalar reverses the arrow. 2a is twice as long as a; -a is the same length the other way.
How it works
- Multiply each component by kThe stretch.
- Flip direction if k is negativeThe sign.
- Keep |k a| = |k| |a|The length.
5Product of Two Vectors
Products: dot a·b = |a||b|cosθ (a number); cross a×b as a vector perpendicular as taught, |a×b|=|a||b|sinθ. Dot is work-like; cross is area-like. 2i·3i=6; 2i·3j=0.
Treating a·b as a vector is a miss.
Figure. The scalar product is |a||b|cosθ. The dashed drop from the tip of a meets b at the projection used in that product.
How it works
- Name dot (scalar) or cross (vector)The product.
- Use cos for dot, sin for |cross| as taughtThe size.
- Keep perpendicular as dot 0A test.
6A definition is a test you can run
Vector-word is a test: magnitude-and-direction, a type, a sum, a stretch, or a product. If you only say “arrow”, you have a heading.
A compass-sticker is not the test.
Figure. A vector is named by its length and its direction. |a| is the measured span of the arrow, not a second object.
How it works
- Name size-and-direction or the operationThe object.
- Give the school sentenceThe test.
- Then the word has contentThe definition ran.
7Name the given before the unknown
The given is two component-writes. The unknown is the sum or the product. Copy 2i+3j before you treat it as 3i+2j.
Using a·b = |a||b| without the angle is a silent skip.
Figure. Write the given shadows first: ax along x, ay along y. Only then ask for |a| or an angle.
How it works
- Copy the components or the angleThe given.
- Name sum, stretch, or productThe unknown.
- Then computeGiven first.
8One worked case is enough at this class
One worked case is enough: (2i+3j)+(4i-j)=6i+2j. Do not stack four crosses. The case teaches component-add.
A poster of ten arrows is not more science.
Figure. Components add house by house: (3, 1) + (1, 2) = (4, 3). The drawn sum matches that pair.
How it works
- Take one sum or one dotThe case.
- Compute on componentsThe teach.
- Stop — one caseThis class.
(2i+3j)+(4i-j) is
- 6i+2j
- 6i+4j
- 8
Matching seats.
Notes
- The official chapter title is “Vector Algebra”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- |a|=sqrt(ax^2+ay^2+az^2)
- a·b=|a||b|cosθ
- |a×b|=|a||b|sinθ
Recap
Hold these pegs from the official chapter “Vector Algebra”. The wording is ExamMaster’s teaching, not a textbook recap.
- Some Basic Concepts
- Basic concepts: a vector has magnitude and direction as taught, written as a directed segment or as components.
- Types of Vectors
- Types: zero, unit, equal, collinear, negative as taught.
- Addition of Vectors
- Addition: triangle or parallelogram as taught; components add matching seats.
- Multiplication of a Vector by a Scalar
- Scalar multiplication: k a stretches (or flips if k<0) as taught.
- Product of Two Vectors
- Products: dot a·b = |a||b|cosθ (a number); cross a×b as a vector perpendicular as taught, |a×b|=|a||b|sinθ.
- A definition is a test you can run
- Vector-word is a test: magnitude-and-direction, a type, a sum, a stretch, or a product.
Practise Vector Algebra
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device