Summary
NCERT Class 12 Maths Chapter 10 covers Vector Algebra, introducing scalars and vectors, types of vectors, vector addition laws, scalar (dot) product, and vector (cross) product with their geometric and algebraic properties.
Chapter 10 of NCERT Class 12 Mathematics Part II introduces Vector Algebra, distinguishing scalar quantities (magnitude only, e.g., speed, mass) from vector quantities (magnitude and direction, e.g., displacement, force). It covers position vectors, direction cosines, types of vectors (zero, unit, coinitial, collinear, equal), triangle and parallelogram laws of vector addition, scalar multiplication, component form of vectors, section formula, scalar (dot) product, projection of a vector on a line, and vector (cross) product. Applications include finding areas of triangles and parallelograms using cross products.
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
This is the complete Mathematics Part II Chapter 10 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all CBSE Class 12 textbooks.
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