5.0 VECTORS 5.2 Vectors in Two and Three Dimensions 5.2 Vectors in Two and Three Dimensions 5.3 Scalar Product 5.4 Vector Product 5.5 Application of Vectors in Geometry 5.1 Introduction to Vectors

5.2 Vectors in Two Dimensions

Learning Outcomes (a) to understand the concept of vectors (b) to discuss the addition of vectors using the triangle laws (c) to define vectors in two dimension

Introduction to vectors Definition A scalar is a quantity that has only magnitude. Example: mass, temperature and volume A vector is a quantity that has both magnitude and direction. Example: velocity, force, weight and momentum

Vectors Representation Geometrically, vectors can be represented by a directed line segment. B A Head(Terminal Point) Tail (Initial Point)

A vector can be written as OR B A AB Magnitude ( also called norm ) is the length of the line AB Direction – The arrow head on the line AB

Types of Vectors Zero Vector (Null Vector) - A vector which has zero magnitude,

AND

Operation on Vectors Triangle Law (a) Addition of Vectors

a +b b + a Parallelogram Law

(b) Scalar Multiplication of Vectors

Vectors in Two Dimensions In xy-plane y x - a unit vector in the positive direction of y-axis - a unit vector in the positive direction of x-axis

Let A (3,4) y x A (3,4) C B O OB = 3, soOB = 3i BA = 4, soBA = 4j By using Triangle Law: Position Vector

i.e the position vector of A(3,4) is Similarly, the position vector of any point A(a 1,a 2 ) is Similarly, the magnitude of any vector

Direction Cosine Of A Vector α β A (3,4) y x C B O

Similarly, the direction cosine of any vector AND

Check

Addition and Subtraction of Vectors

NOTES

Scalar Multiplication

Example 1

y x B A O Solution: A ( 3, -1 ) B ( -2,3 )

Example 2 Given A (5,1) and B (2,-3). Find: (i) position vectors of A( ) and B( ) (ii)| + | and a unit vector of + (iii)Vector and its direction cosines

Solution:

Triangle Law Position vectors

CONCLUSION The position vector of any point P(a 1,a 2 ) is The magnitude of any vector

The direction cosine of any vectorAND Using position vectors