Vectors: Magnitude and directionExamples for Vectors: force – acceleration- displacement

Scalars: Only Magnitude A scalar quantity has a single value with an appropriate unit and has no direction.

Examples for Scalars: mass- speed- work-Distance- Energy-Work-Pressure

Motion of a particle from A to B along an arbitrary path (dotted line). Displacement is a vector

2.1 :An introduction to vectors

Vectors:

• Represented by arrows (example displacement).

• Tip points away from the starting point.

• Length of the arrow represents the magnitude

• In text: a vector is often represented in bold face (A) or by an arrow over the letter.

• In text: Magnitude is written as A or A

A

This four vectors are equal because they have the same magnitude and same length

Adding vectors:

Draw vector A. Draw vector B starting at the tip of vector A.

The resultant vector R = A + B is drawn from the tail of A to the tip of B.

Graphical method (triangle method):

Two vectors can be added using these method:1- tip to tail method.2- the parallelogram method.

1- tip to tail method.

5

Adding several vectors together.

Resultant vector

R=A+B+C+D

is drawn from the tail of the first vector to the tip of the last vector.

6

A + B = B + A(Parallelogram rule of addition)

Commutative Law of vector addition2- the parallelogram method.

Associative Law of vector addition

A+(B+C) = (A+B)+C

The order in which vectors are added together does not matter.

Negative of a vector.The vectors A and –A have the same magnitude but opposite directions. A + (-A) = 0

A -A

Subtracting vectors:

A - B = A + (-B)

Multiplying a vector by a scalarThe product mA is a vector that has the same direction as A and magnitude mA.

The product –mA is a vector that has the opposite direction of A and magnitude mA.

Examples: 5A; -1/3A

• Given , what is ?

s

3s

s

s

s

s

s

s

Components of a vector

sin

cos

AA

AA

y

x

22yx AAA

x

y

A

A1tan

The x- and y-components of a vector:

The magnitude of a vector:

The angle between vector and x-axis:

The signs of the components Ax and Ay depend on the angle and they can be positive or negative.

Examples)

Unit vectors• A unit vector is a dimensionless vector having a magnitude 1.• Unit vectors are used to indicate a direction. • i, j, k represent unit vectors along the x-, y- and z- direction• i, j, k form a right-handed coordinate system

The unit vector notation for the vector A is:

OR in even better shorthand notation:

,x yA AA

ˆ ˆx yA i A j A

• A unit vector is a dimensionless vector having a magnitude 1.• Unit vectors are used to indicate a direction. • i, j, k represent unit vectors along the x-, y- and z- direction• i, j, k form a right-handed coordinate system

Adding Vectors by Components

We want to calculate: R = A + B

From diagram: R = (Axi + Ayj) + (Bxi + Byj)

R = (Ax + Bx)i + (Ay + By)j

The components of R:Rx = Ax + Bx

Ry = Ay + By

2222 )()( yyxxyx BABARRR

The magnitude of a R:

xx

yy

x

y

BA

BA

R

RtanThe angle between vector R and x-axis:

example

Example A force of 800 N is exerted on a bolt A as

show in Figure (a). Determine the horizontal and vertical components of the force.

The vector components of F are thus,

and we can write F in the form

Example :The angle between where

with the positive x is:

1. 61°2. 29°3. 151°4. 209°5. 241°

A

A x

A y

Ax 25 & Ay 45

Example :

Example :F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S

F=F1+F2+F3

W

Ex : 2 – 10 A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?

example

Answer is d