Chapter 3 Vectors

In Physics we have parameters that can be completely described by a number and are known as “scalars” .Temperature, and mass are such parameters

Other physical parameters require additional information about direction and are known as “vectors” . Examples of vectors are displacement, velocity and acceleration.

In this chapter we learn the basic mathematical language to describe vectors. In particular we will learn the following:

Geometric vector addition and subtraction Resolving a vector into its components The notion of a unit vector Add and subtract vectors by components Multiplication of a vector by a scalar The scalar (dot) product of two vectors The vector (cross) product of two vectors

(3-1)

An example of a vector is the displacement vector which describes the change in position of an object as it moves from point A to point B. This is represented by an arrow that points from point A to point B. The length of the arrow is proportional to the displacement magnitude. The direction of the arrow indicated the displacement direction.

The three arrows from A to B, from A' to B', and from A'' to B'', have the same magnitude and direction. A vector can be shifted without changing its value if its length and direction are not changed.

In books vectors are written in two ways: Method 1: (using an arrow above)

Method 2: a (using bold face print)

The magnitude of the vector is indicated by italic print: a

a

(3-2)

Geometric vector Addition

Sketch vector using an appropriate scale

Sketch vector using the same scale

Place the tail of at the tip of The vector starts from the tail of

and terminates at the tip of

a

b

b as a

s a b

commutative

Negative of a given

Vector addition is

has the same magnitude

vecto

as but opposite direction

r b b

b

a b b a

b b

(3-3)

Geometric vector Subtraction

We write:

From vector

Then we add to vector

We thus reduce vector subtraction to vector addition which we know how to do

we find

d a b a b

b b

b

b

a

d a

Note: We can add and subtract vectors using the method of components. For many applications this is a more convenient method

(3-4)

A B

C

A component of a vector along an axis is the projection of the vector on this axis. For example is the projection of along the x-axis. The component is defined by drawing straight lines fr

x

x

aa a

om the tail and tip of the vector which are perpendicular to the x-axis. From triangle ABC the x- and y-components of vector are given by the

cos , equations:

If we know

in

sx ya a a

a

aa

a

2 2

and we can determine and .

From triangle ABC we h

, tan

ave:

yx

x

yx

y

aa a

a

a a

a

(3-5)

Unit Vectors

A unit vector is defined as vector that has magnitude equal to 1 and points in a particular direction.

A unit vector is defined as vector that has magnitude equal to 1 and points in a particular direction. Unit vector lack units and their sole purpose is to point in a particular direction. The unit vectors along the x, y, and z axes

are labeled , , and , respectˆˆ y. lˆ ivei j k

Unit vectors are used to express other vectorsFor example vector can be written as:

. ˆ ˆThe quantities and are called

the of vec

ˆ ˆ

vect toror component s

x

x y

ya a i

a

a i a j

a

a j

(3-6)

x

O

y

a

r b

Adding Vectors by Components

ˆ ˆ ˆ ˆWe are given two vectors and ˆ ˆWe want to calculate the vector sum

The components and are given by the equations:

and

x y x y

x y

x y

x x x y y yr a b r a b

a a i a j b b i b j

r r i r j

r r

(3-7)

x

O

y

a

d

b Subtracting Vectors by Components

ˆ ˆ ˆ ˆWe are given two vectors and

We want to calculate the vector difference ˆ ˆ

The components and of are given by the equations:

and

x y x y

x y

x y

x x x

a a i a j b b i b j

d a b d i d j

d d

a b

d

d

y y yd a b

(3-8)

Multiplication of a vector by a scalar r esults in a new vector The magnitude of the new vector is given by

Multiplying a Vecto

:

If 0 vector has the

r by a Scalar

| |b sa s b sa

ab

s b

The S

same

calar

direction

Product o

as vector

If 0

f two

vector has a direction opposite to that of vector

The scalar product of two vectors and is given by:

Vectors

=

a

s b a

a b a

a b

b

The scalar product of two vectors is also known as the product. The scalar product in termsof vector components is given by the equation:

cos "dot"

= x x y y z z

ab

a b a b a b a b

(3-9)

The vector product of the vectors and is a vector The magnitude of is given

The Vector Pro

by the equation:

The direction of is perpend

duct of two Vectors

sicular

in

c a b a bc

cc

cab

right hand

to the plane P defined

by the vectors and The sense of the vector is given by the :

a. Place the vectors and tail to tailb. Rotate in the plane P along

rule

the shortest an

a bc

a ba

gle

so that it coincides with c. Rotate the fingers of the right hand in the same directiond. The thumb of the right hand gives the sense of The vector product of two vectors is also known as

b

c

"crossthe p" roduct(3-10)

The vector components of vector are given by the equations:

,

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ,

The Vector Product in terms of Vector Components

,

x y z z y

x y z x y z x y z

c a b

c a b a b

a a i a j a k b b i b j b k c c i c j c kc

Note: Those familiar with the use of determinants can use the expres

,

Note: The order of the two vectors in the cross product is importa

sion

n

c c

t

y z x x

x y z

x

z z x

y z

y y x

i j ka b a a a

b b b

a b a b a b a b

b

a a b

(3-11)