Chapter 8 VECTORS Review

Chapter 8 VECTORS R

eview

1. The length of the line represents the magnitude and the arrow indicates the direction.2. The magnitude and direction of the vector is

clearly labeled.

A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

Two vectors are equal if they have the same direction and magnitude (length).

Blue and orange vectors have same magnitude but different direction.

Blue and green vectors have same direction but different magnitude.

Blue and purple vectors have same magnitude and direction so they are equal.

P

Q

Initial Point

Terminal Point

magnitu

de is th

e length

direction is

this angle

How can we find the magnitude if we have the initial point and the terminal point?

22 , yx

11, yx

The distance formula

How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

The speed of an airplane relative to the ground is affected by wind. (actual speed w/respect to the ground

If we know the speed and direction of the plane and the speed and direction of the wind, we can calculate the resulting speed and direction of the airplane.

22 , yx

11, yx

Q

Terminal Point

direction is

this angle

Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y).

yx,

0,0If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.

P

Initial Point

A vector whose initial point is the origin is called a position vector

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

v w

Initial point of vv

w

Move w over keeping the magnitude and direction the same.



Terminal point of w

w

The negative of a vector is just a vector going the opposite way.

v

v

A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

vv

vv3

uv

w vu u

vw3w w

w

Using the vectors shown, find the following:

vu u

v

vwu 32

uu w

w wv

0o

N

180o

270o

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S



E

N

W

S



200 mph

50 mph

VR = 165 mph @ 40° N of E

Use a metric ruler and protractor to find each sum or difference.Then find the magnitude and amplitude of each resultant.

€

va

Magnitude = 2 cmAmplitude = 60°

€

vb

€

vc



1.) 2.)

€

va +

v b

€

vb −

v c

60°

140°

€

va +

v b

2 cm

3 cm

3.9 cm

140°

130°

€

vb −

v c

3 cm

1 cm

4 cm

110°

139°

Use a metric ruler and protractor to find each sum or difference.Then find the magnitude and amplitude of each resultant.

€

va


€

vb

€

vc



1.) 2.)

€

2v a +

v b

€

vb + 3

v c

60°

140°

€

2v a +

v b

2 cm

3 cm

5.5 cm

140°310°

€

vb + 3

v c

3 cm1 cm

0.6 cm93°

217°

2 cm

FIND THE MAGNITUDE OF THE VERTICAL AND HORIZONTAL COMPONENTS OF EACH VECTOR.

€

va


€

vb

€

vc



x x

x

y y y

€

sin60° =y

2

€

cos60° =x

2

y = 1.7 cm

x = 1 cm

€

sin40° =y

3

€

cos40° =x

3

y = 1.9 cm

x = -2.3 cm €

sin50° =y

1

€

cos50° =x

1

y = -0.7 cm

x = 0.6 cm

€

v = a,b( )

Vectors are denoted with bold letters

(a, b)

This is the notation for a position vector. This means the point (a, b) is the terminal point and the initial point is the origin.

We use vectors that are only 1 unit long to build position vectors. i is a vector 1 unit long in the x direction and j is a vector 1 unit long in the y direction.

ij

€

v = a,b( ) = ai + bj

(3, 2)

ij

€

v = 3,2( ) i i

j

jiv 23

UNIT VECTORS

jiji 4352

If we want to add vectors that are in the form ai + bj, we can just add the i components and then the j components.

jiv 52

wv ji

Let's look at this geometrically:

i2

j5 v

i3

j4w

ij

When we want to know the magnitude of the vector (remember this is the length) we denote it

v 22 52 Can you see from this picture how to find the length of v?

29

jiw 43

A unit vector is a vector with magnitude 1.

jiw 43

If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value.

What is ?w

2 23 4 w 525

If we want to find the unit vector having the same direction as w we need to divide w by 5.

jiu5

4

5

3

Let's check this to see if it really is 1 unit long.

2 23 4 25

15 5 25

u

FIND THE ORDERD PAIR THAT REPRESENTS THE VECTOR “A” TO “B”. THEN FIND THE MAGNITUDE OF “AB”

1.) A(2, 4), B(-1, 3) 2.) A(4, -2), B(5, -5)

3.) A(-3, -6), B(8, -1)

€

vA

v B = (−1 − 2,3 − 4)

= (−3,−1)

v A

v B = (−3)2 + (−1)2

= 10

€

vA

v B = (5 − 4,−5 − −2)

= (1,−3)

v A

v B = (1)2 + (−3)2

= 10

€

vA

v B = (8 − −3,−1− −6)

= (11,5)

v A

v B = (11)2 + (5)2

= 146

FIND THE SUM OR DIFFERENCE OF THE GIVEN VECTORS ALGEBRAICALLY. EXPRESS THE RESULT AS AN ORDERED PAIR.

1.) 2.)

€

2v i + 6

v j

€

4v i − 5

v j

(2, 6)(4, -5)

FIND AN ORDERED PAIR TO REPRESENT VECTOR “U”

IF:

€

vv = (2,−1)v w = (−3,5)

1.) 2.) 3.)

4.) 5.) 6.)

€

vu =

v v +

v w

€

vu =

v v −

v w

€

vu = 3

v v

€

vu =

v w − 2

v v

€

vu = 2

v v + 3

v w

€

vu = 5

v w − 2

v v

(-1, 4) (5, -6) (6, -3)

(-7, 7) (-5, 13) (-19, 27)

Chapter 8 VECTORS Review

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