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Vectors Quantities that have magnitude but not direction are called scalars Ex Area volume temperature time etc Quantities such as force acceleration velocity or displacement that ID: 510219 Download Presentation

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direction vector airplane east vector direction east airplane vectors wind due magnitude called standard speed north ground mph acquires

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Slide1

12.2

Quantities that have

magnitude but not direction are called scalars.Ex: Area, volume, temperature, time, etc.

Quantities such as force, acceleration, velocity or displacement that have direction as well as magnitude are represented by directed line segments, called vectors.

A

B

initial

point

terminalpoint

The

length

of the vector is calledthe magnitude and is denoted by

DefinitionsSlide3

A

vector is in standard position if the initial point is at the origin.

x

y

The

component form

of this vector is:

Vectors

are

equivalent

if they have the same length and direction.

then the

component form of

is:

If

are initial and terminal points of a vector,

P

Q

(

c,d

)

(

a,b

)

v

(a-c, b-d)

xSlide4

P

Q

(-3,4)

(-5,2)

The component form of

is:

v

(-2,-2)

The

magnitude is

ExampleSlide5

i

and

j

If

then

v

is a

zero vector

:

are called the

standard basis vectors

.

The

magnitude

of

is:

If

then

v

is a

unit vector

.Slide6

and

If

then

v

is a

zero vector

:

are called the

standard basis vectors

.

The

magnitude

of

is:

If

then

v

is a

unit vector

.

Vectors in Space Slide7

Vector sum:

Vector difference

Scalar Multiplication

:

Negative

(opposite):

Vector

v

is parallel to

u

if and only if

v =

k

u

for some

k.

Vector OperationsSlide8

v

v

u

u

u+v

u

+

v

is the

resultant vector

.

v

v

u

u

u-v

u

-

v

is the

resultant vector

.

Parallelogram LawSlide9

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60

o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

ApplicationSlide10

A Boeing 727 airplane, flying due

east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

uSlide11

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a

70-mph tail wind acting in the direction of 60o north of east

. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

60

oSlide12

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60

o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

We need to find the magnitude and direction of the

resultant vector

u + v

.

u+vSlide13

N

E

v

u

The component forms of

u

and

v

are:

u+v

500

70

Therefore:

and:

The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5

o

north of east.Slide14

i

j

v

is called a

linear combination

of i and

j

Any vectors can be written uniquely

in terms of standard basis vectors :

v

(measured counterclockwise) with the positive

x

-axis

then

v

can be written as

If

v

is

any nonzero vector that makes an angle

Linear CombinationSlide15

v

is called a

linear combination of

i, j and k

Standard basis vector notation

Linear Combination in SpaceSlide16

1) Find the unit vector in the direction of

v

2) Determine whether the points are collinear:

3) Show that the following points form the vertices of a parallelogram:

Examples

Shom More.... By: marina-yarberry
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