VECTORS

A bee will fly from rose to roseTo gather up some nectar.

And though his path is quite complexHis displacement is a vector!

WHAT IS A VECTOR?

A vector is a device used when the direction of the parameter being measured is important.Some examples of vectors are: velocity, force, momentum, electric field strength, and torqueA vector quantity is typically written in boldface or by placing an arrow over the label.A vector quantity may be represented by drawing an arrow. The length of the arrow is proportional to the magnitude of the vector quantity and the arrow points in the direction of the vector

WHAT IS A VECTOR?

The arrow is usually drawn at the origin of a coordinate system and the angle is measured (in radians or degrees) from an axis.Once a vector is drawn, it may be moved to any other location provided the length and direction of the arrow does not change

VECTOR ADDITION(determining a resultant)

The maximum resultant occurs when the angle between the vectors being added is 00 and the minimum when the angle is 1800

Vectors may be added several different ways: 1. Graphically – draw all vector quantities to

scale, arrange head to tail, connect the origin of the first vector to the head of the last. (REMEMBER THE BEE)

2. Mathematically Law of Cosines - C2= A2 + B2 – 2ABcos q (The

Theory of Pythagoras is a special case) Components (Two and three dimensions coming

soon!)

DETERMINING COMPONENTSDetermine the components of a velocity vector with magnitude 20 m/s directed at 37o above the + x-axis.

v =20m/s

37o

vy

vx

s

m

s

mv

s

m

s

mv

y

x

1237sin20

16)37cos(20

EXAMPLE OF VECTOR ADDITION

A

B

C

Cx Ax BxCy Ay By

Ay

Ax Cx

CyBx

By

VECTOR SUBTRACTION

To determine the difference between two vectors arrange them with the tails togetherThe direction of the resultant is determined by the order of the equation

A

B A - B B - A

A

B

UNIT VECTORSThis is a fancy name for a vector of

length one (1).A unit vector is constructed by

dividing a vector by its magnitude

kji ˆ,ˆ,ˆ

jiA ˆ3ˆ6

Unit vectors in the x, y, z direction are denoted as:

A unit vector with an x component of 6, a y component of –3, and a z component of zero is written as:

A

AA

ˆ

UNIT VECTORS - example

If: jiA ˆ3ˆ6

Then the magnitude of A is 6.71 and the unit vector is:

jiA ˆ45.ˆ89.ˆ

22 36 A

^^^

71.6

3

71.6

6jiA

ADDITION IN UNITVECTOR NOTATION

Adding vectors this way is easy!

kjiA ˆ3ˆ4ˆ8

jiB ˆ5ˆ2

kjiBAC ˆ3ˆˆ6

OTHER FUNCTIONS WITH VECTORS

Dot product (scalar product) Used to determine the amount of one vector in the direction of another or the angle between vectorsWork is an example of a dot product

(W= F.d (“f dot d”) or, Fdcosq)

A

B

cosABBA

q yyxx BABABA

Using unit vector notation:

Dot product example

Determine the angle between the following two vectors:

jiB

jiA

ˆ9ˆ2

ˆ4ˆ7

85

65

B

A

yyxx BABABA

0

1

7.47

)85)(65(

50cos

cos)85)(65(50

50)9)(4()2)(7(

Cross product (vector product) Used to produce a third vector.The result is perpendicular to the first two vectors (See fig. 3 – 20)Torque is an example of a cross product

(t=r x F (“r cross F”) or rFsinq)

OTHER FUNCTIONS WITH VECTORS

A

B

qsinABC

CBA

Calculating a cross product using a

determinant

This method will require practice!!See TACTIC 5 ON PAGE 51.

YX

YX

ZX

ZX

ZY

ZY

ZYX

ZYX BB

AAk

BB

AAj

BB

AAi

BBB

AAA

kji

BA ˆˆˆ

ˆˆˆ

kABBAjABBAiABBA YXYXXZXZZYZYˆ)(ˆ)(ˆ)(

DOT AND CROSS PRODUCTS

http://www.tutorvista.com/content/physics/physics-iii/kinematics/scalar-product-animation.php

VECTORS!KEEP THEM STRAIGHT AND YOUR LIFE WILL

HAVE DIRECTION!