Summary
Chapter 10 of the Class 12 Maths NCERT textbook, "Vector Algebra", introduces scalars and vectors, types of vectors, vector addition laws, scalar (dot) product, and vector (cross) product with their geometric and algebraic properties.
- Scalars versus vectors — The chapter draws the key distinction between quantities that need only a magnitude (speed, mass) and those that also carry direction (displacement, force), then classifies vectors — zero, unit, collinear, equal — and represents them through position vectors and direction cosines.
- Adding and scaling vectors — Vectors combine geometrically through the triangle and parallelogram laws and algebraically in component form. Scalar multiplication stretches or reverses them, and the section formula locates points dividing a segment — building the machinery for the products that follow.
- Two kinds of vector products — The dot product measures alignment and detects perpendicular vectors, while the cross product yields a new vector perpendicular to both. Their geometry directly gives areas of triangles and parallelograms, linking vector algebra back to measurement.
Key points & formulas
- 01A vector is a directed line segment with both magnitude and direction; its magnitude is always non-negative
- 02Vector addition follows the Triangle Law (AC = AB + BC) and the equivalent Parallelogram Law (diagonal represents the resultant of two adjacent-side vectors)
- 03The scalar (dot) product of two vectors a and b is defined as a·b = |a||b|cosθ; two nonzero vectors are perpendicular if and only if their dot product is zero
- 04For vectors in component form, the dot product equals a1b1 + a2b2 + a3b3, and the cross product is computed using a 3×3 determinant with unit vectors i, j, k in the first row
- 05The cross product a×b gives a vector perpendicular to both a and b with magnitude |a||b|sinθ; it is not commutative (b×a = −a×b)
- 06The area of a triangle with adjacent sides represented by vectors a and b equals (1/2)|a×b|, and the area of a parallelogram equals |a×b|
Frequently asked questions
01What is the difference between scalar and vector quantities?
Scalar quantities have only magnitude (e.g., length, mass, speed, temperature), while vector quantities have both magnitude and direction (e.g., displacement, velocity, acceleration, force). For example, '1.6 metres tall' is a scalar, but 'kick the ball north-east with force F' involves a vector.
02What are the two laws of vector addition covered in Chapter 10?
The Triangle Law states that if a girl moves from A to B and then B to C, the net displacement AC = AB + BC. The Parallelogram Law states that if two vectors are represented by adjacent sides of a parallelogram, their sum is represented by the diagonal through their common point. Both laws are equivalent to each other.
03How is the cross product of two vectors used to find the area of a triangle?
If two sides of a triangle are represented by vectors a and b, the area equals (1/2)|a×b|. For example, for triangle with vertices A(1,1,1), B(1,2,3), C(2,3,1), computing AB×AC gives −4i+2j−k with magnitude √21, so the area is (1/2)√21.
04Is the NCERT Class 12 Maths Chapter 10 PDF free to download?
Yes, the NCERT Class 12 Maths Part II Chapter 10 (Vector Algebra) PDF is completely free to download on cbseprepmaster.com.
More chapters in Mathematics Part II
Read Chapter 10 of Mathematics Part II, the Class 12 Mathematics NCERT textbook (2026-27 edition), online for free: the complete chapter as published by NCERT with every diagram, solved example and exercise, with step-by-step solutions, answers and revision notes. Open the NCERT PDF above, or browse all CBSE Class 12 textbooks.
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