# Vector Algebra Notes, Class 12 Mathematics | SwaVid Vector Algebra, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/vector-algebra Source: https://www.swavid.com/cbse/class-12/maths/chapter/vector-algebra # Vector Algebra ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 12 Mathematics on SwaVid ### Basic concepts and types of vectors ### Addition and scalar multiplication ### Position vector and section formula ### Scalar product of two vectors ### Vector product of two vectors ### Next chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 12 › Mathematics › Vector Algebra Notes written by SwaVid from the NCERT textbook. Class 12 Mathematics, Chapter 10. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter introduces vectors as directed line segments with both magnitude and direction, covering their algebraic properties and geometric representations. Students learn to compute position vectors, scalar products, and vector products to solve problems involving angles, projections, and geometric areas. Source: NCERT Mathematics Part II (Reprint 2026-27) , Chapter 10, Vector Algebra A vector is a quantity that has magnitude as well as direction. Standard types include zero vectors, unit vectors, collinear vectors, equal vectors, and negative vectors. Vectors are added using the triangle or parallelogram law of vector addition. Multiplying a vector by a scalar λ changes its magnitude by |λ| and reverses its direction if λ is negative. The position vector of a point P(x, y, z) relative to the origin is r = x î + y ĵ + z k̂ with magnitude √(x² + y² + z²). The section formula finds the position vector of a point dividing a line segment internally or externally in a given ratio. The dot product of two vectors a and b is a · b = |a| |b| cos θ, producing a real number. Two non-zero vectors are perpendicular if and only if their scalar product equals zero. The cross product of two vectors a and b is a vector perpendicular to both, given by a × b = |a| |b| sin θ n̂. Two non-zero vectors are collinear if and only if their vector product is the zero vector. Try each question first. Then tap it to see the answer. Calculate the magnitude: |a| = √(2² + 3² + 1²) = √(4 + 9 + 1) = √14 Divide the vector by its magnitude: â = a / |a| = (2 î + 3 ĵ + k̂) / √14 Write in component form: (2/√14) î + (3/√14) ĵ + (1/√14) k̂ (2/√14) î + (3/√14) ĵ + (1/√14) k̂ Compute a · b = (1)(7) + (3)(-1) + (7)(8) = 7 - 3 + 56 = 60 Compute |b| = √(7² + (-1)² + 8²) = √(49 + 1 + 64) = √114 Projection of a on b = (a · b) / |b| = 60 / √114 60 / √114 Compute a × b = î(1·1 - 4·(-1)) - ĵ(3·1 - 4·1) + k̂(3·(-1) - 1·1) Simplify components: a × b = î(1 + 4) - ĵ(3 - 4) + k̂(-3 - 1) = 5 î + ĵ - 4 k̂ Compute the magnitude: |a × b| = √(5² + 1² + (-4)²) = √(25 + 1 + 16) = √42 √42 square units Three Dimensional Geometry . Determinants . The 13 chapters of Class 12 Mathematics . CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Three Dimensional Geometry. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 10 NCERT book - 80 marks, theory paper - 3 quick checks - Treating dot product and cross product interchangeably or assuming a × b is commutative . Remember that dot product yields a scalar while cross product yields a vector. Cross product is anti-commutative, meaning a × b = -(b × a). - Dividing one vector by another vector . Division by a vector is not defined in vector algebra. Express relations using scalar multiples, dot products, or cross products instead. - Forgetting the unit vector n̂ in the definition of the cross product . The expression |a| |b| sin θ gives only the magnitude of the cross product. The complete product is a vector: a × b = |a| |b| sin θ n̂. ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 12](https://www.swavid.com/cbse/class-12) - [Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 13 chapters](https://www.swavid.com/cbse/class-12/maths/chapters) - [NCERT Mathematics Part II (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/three-dimensional-geometry) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/determinants) - [Class 12 Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup)