Vector Unit

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VECTOR UNIT

GSS Science


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Introduction

In this unit we will study all aspects of vectors,including vector addition equations.

We will then use the knowledge gained tosolve two dimensional vector problems suchas plane flights and force problems.

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Vector Unit:

Objectives

Define a vector

Identify resultant and component vectors on a diagram

Write a vector equation Use graphical methods to resolve a vector into two

perpendicular vectors

Use trig to add/subtract vectors

Resolve vectors into component vectors to add/subtract them Use cosine and sine law to add/subtract vectors

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Vectors

Two types of quantities exist in the world: Scalers a quantity with a value but without

direction i.e. speed = 12 m/s

Vectors a quantity with magnitude (value)and a direction (heading) i.e. velocity = 12 m/sdue north

Scalersinclude: Speed, energy, time, work, distance, mass

Vectors include: Velocity, acceleration, force, momentum,

displacement

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Vector Direction

Lets learn about directions for vectors.

We use a compass for basic direction/headings

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N

E

S

W


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Vector Direction

If a vector is heading along one of the main heading (such asN, E, S, or W) just give it that direction

i.e. A train has a velocity of24 m/s (S)

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N

24 m/s


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Vector Direction

If the vector is exactly mid-way between two main headingsthen we can use:

45o N of W or

NW, SW, SE, SW as the direction

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NNWNE

SW SE

45o

45o

W E

S


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Vector Direction

But what about other directions? In this case we use a direction from one of the main headings (N, E, S,

or W)

This vector would become: 24 m/s at 32o N of E

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N

E

S

W

24 m/s

32o

The vector direction 32o

above the main EAST

direction. Hence we say32o N of E.


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Vector Direction

There is another way of indicating this direction In this case we use a direction measured away

from the north direction

This vector would become: 24 m/s at 58o E of N

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N

E

S

W

24 m/s

58o

The vector direction 58o from

the main North direction.Hence we say 58o E of N.


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Vector Direction

Every vector has two ways of indicating directions

You only need to state one of them

For example: 65 m at 22o N of W or 65 mat 68o W of N (same direction)

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N

E

S

W

65 m

22o

68o

Remember, you only need

one direction!

Note: The two directions must

always add up to 90o

. Why?


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Vector Direction

Now try these: State the magnitude of each vector and give bothpossible directions

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35 N N

24o

N

45 m

13o

Vector = Vector =

35 N at 240 N of W

35 N at 66o W of N

45 m at 13o S of E

45 m at 77o E of S


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Vector Direction

Bearing another direction system used byairplanes and ships for navigation

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N

0

o

90o

180o

270o


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Vector Direction

This is easy: 24 Km bearing 125o (90o + 35o)

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N0o

90o

180o

270o 35o

24 Km bearing 125o


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Vector Direction

Now try these:

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N

0o

180o

250 km

45o

60o

210 km

225 km

250 km bearing 45o

210 km bearing 150o

225 km bearing 270o


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Adding & Subtracting Vectors

Graphing method:

Vectors in the same or opposite direction

Notes from Over-head

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When vectors are not in same or oppositedirection:

A little more work is involved:

You need a ruler & a protractor Over Head Notes

Do Graphing Vectors Worksheet

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Vector equations:

What does a + b =R really mean?

What does a - b=R really mean?

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Take these two vectors:

ab

a + b = R

R

(adding Vectors)


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Take the same two vectors:

ab

a - b = R

What does this mean?

Remember your math: a - b = a + (- b)In vectors (b) and (-b) vectors are related: They have the same

magnitude but have opposite directions!

a

- ba - b = a + (-b) = R

R

(Subtracting Vectors)

ddi b i


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Take the another two vectors: c

d

c + d = R c - d = R

c

R

c

d

- d

R

ddi & b i


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Take the another two vectors: e

e + f = R- e + f = R

R

R

e

f

e

f

f

ddi i


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Vector Addition Component Method

Navigational vector problems are usually

solved by graphing, if you are a little offcourse no problem the airport has to bearound here somewhere!

But building bridges or airplanes need a moreprecision to adding vectors in solvingconstruction problems.

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t dditi C t th d


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Imagine this conversation betweentwo engineers:

Engineer #1 - Well the graphing

method of adding vectors gave me anapproximate load limit for the bridge.I guess it was a little low. You bettercall in emergency vehicles and

cranes Engineer #2 - I guess so I told you

we should have used the componentmethod!

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V t Additi


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Any vector can be re-made into two orthonganol (right-angled) vectors

These are called component vectors

This is some vector a:

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a

26o

V t Additi C t M th d


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This is the horizontal component vector (ax)

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a

26o

ax

V t Additi


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This is the vertical component vector (ay)

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a

26o

ay

V t Additi C t M th d


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These two vector can be shown as the addition of two right-angled vectors

One is the horizontal component (ax)

One is the vertical component (ay)

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Head to Tail

Method

a

26o

ax

ay



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The two vectors are sometimes shown in tail totail addition method

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a

26o

ax

ayTail to Tail

Method



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In summary there are two ways to show the addition of vectorsand they both (of course) end up with the same resultant vector

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a

26o

ax

ay

Tail to Tail Method

a

26o

ax

ay

Head to Tail Method

or



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Combing the two methods results in parallelogram!! It doesnt matter which way it is shown, you end up at the same

end-point!

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a

26o

ay

ax

26o

There are two identical

triangles here.

ax

ay

Head to TailTail to Tail

End point!



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You have to use a little trig. (sine, cosine and tangent) towork with these vector components

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a

26o

ay = a x sin 26o

ax = a x cos 26o



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Finding resultant vector: Using some basic trigonometry

Sine, Cosine, and Tangent

Right angle triangles

Pythagorean Theory

Over Head Notes

Go on to the Vector Component Worksheet

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Vector Addition Cosine Law


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This method works when the componentvectors are not at right angles to each other.

You have to use two other trigonometryLaws:

Cosine law to find magnitude

Sine Law to find direction

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Examine the following two vectors:

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b

c

These two vectors are not at right angles to one another.

So how can we add these two vectors?



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To find the resultant vector R we use thecosine law:

R = (b2 + c2 (2 x b x c x cos )

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R

b

c



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To use the cosine law you need to know the angle opposite tothe resultant vector that you want to find:

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R

b

c

This will allow you to find the magnitude of the resultant

vector, R, by using the cosine law.

Opposite angle

to the resultant



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To find the direction of the resultant vector usethe sine law:

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Sin B Sin C Sin= =b c

b

c

R

B

C

R



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Over Head Notes

Do the Vector: Cosine Law Problem Worksheet

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Concluding Statements

You now should have a good grasp of dealingwith vectors.

We will be using these pesky vectorsthroughout the Physics 12 course.

Make sure you understand these topics andthat you are fairly comfortable with adding

vectors.

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The End

of this Topic.

On to Vector Kinematics!

Vector Unit

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Transcript of Vector Unit