NCERT Solutions for Class 12 Mathematics Chapter 10: Vectors – Free PDF Download

Chapter 10 of Class 12 Maths is Vectors, covering vector representation, algebra, dot and cross products, and their geometric applications. These Class 12 Mathematics Chapter 10 solutions are also useful as quick revision notes before exams.

Vector Basics

A vector has both magnitude and direction, denoted a or with an arrow. Position vectors locate points relative to the origin. Unit vectors i, j, k represent the coordinate axes.

Types of Vectors

Zero vector (magnitude 0), unit vector (magnitude 1), collinear vectors (parallel to the same line), equal vectors (same magnitude and direction), negative vector (same magnitude, opposite direction).

Algebra of Vectors

Addition follows the triangle/parallelogram law. Scalar multiplication scales magnitude, preserving or reversing direction based on sign. A vector a=a₁i+a₂j+a₃k has magnitude |a|=√(a₁²+a₂²+a₃²).

Section Formula

The position vector of a point dividing the line joining points with position vectors a and b in ratio m:n is (mb+na)/(m+n).

Dot (Scalar) Product

a·b=|a||b|cosθ=a₁b₁+a₂b₂+a₃b₃. Used to find the angle between vectors and to check perpendicularity (a·b=0).

Cross (Vector) Product

a×b=|a||b|sinθ &ncirc; (a vector perpendicular to both), computed via a 3×3 determinant with i,j,k. Used to find area of a triangle/parallelogram and to check parallelism (a×b=0).

Frequently Asked Questions

What is the difference between dot and cross product?
Dot product gives a scalar (related to cosθ); cross product gives a vector (related to sinθ, perpendicular to both vectors).

How do you check if two vectors are perpendicular?
Their dot product equals zero.

How is the area of a triangle found using vectors?
Area=½|a×b|, where a,b are two sides as vectors.

Written by Satish

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