} Represent physical concepts } Include magnitude and direction } Shown as arrows }  A B

} Examples: velocity, displacement, acceleration, force

}  Vectors can be drawn to scale and should AT LEAST show relative magnitude!

(Read this slide Only)

AP Chapter 3 Vectors

• Pick an origin to start your vector diagram

• All vectors have an amount and a direction in degrees or radians!

A 25

• Two or more vectors always add head to tail to yield the resultant vector= with its direction!

A

B

€

θ€

R

Head to Tail

} Attach the tail of each vector to the head of the previous one

} The resultant is drawn from the tail of the first to the head of the last

-A

Rnet

Step 1 Step 2 =

(Draw this) B

C

}  Attach the tail of each vector to the head of the previous one }  The resultant is drawn from the tail of the first to the head of the last

F1

F3 F2

Fnet

Any order works! Fnet

Example 1

Same thing!!

(Draw this)

0

90

180

270

*Always measure from the horizon and speak it as going from the vertical - horizon!

This vector is 20N going 70 North- East

This vector is 20N going 10 SouthWest

1.Always Remember your coordinate axes !

(Draw this)

Component Solving

+x

-y

+y

-x

(Draw this)

Component Solving

+x

-y

+y

+x

2.Put in your vector from its origin with it’s angle from the origin!

€

θ

3.Remember your equations for sin -cos -tan!

Ax = A cos

Ay = A sin

€

θ

θ“x” Component A

“y” Component of A (Draw this)

€

θAx

Ay A

}  A paper airplane is thrown to fly north at 2 m/s. When the wind begins to blow east at 3 m/s, what are the new magnitude and direction of its velocity?

1.  Choose a coordinate system

2.  Choose a scale (I.e. 1 inch = 1 m/s)

3. Redraw head to tail and measure resultant and angle w /ruler + protractor

} A paper airplane is thrown to fly north at 2 m/s. When the wind begins to blow east at 3 m/s, what are the new magnitude and direction of its velocity?

NW E

S Vplane = 2 m/s

Vair = 3 m/s 33.5°

Airplane now moves at 3.6 m/s, 33.5° north of east.

a

θ

c

b

sin θ = opposite = a hypotenuse c

cos θ = adjacent = b hypotenuse c

tan θ = opposite = a adjacent b

To remember: SOH CAH TOA

a

θ

c

b

Pythagorean theorem:

a2 + b2 = c2

c2 = a2 + b2 - 2ab cos C

asinA

=b

sinB=

csinC

A

B

C

c a

b

Easy Trigonometry if you forgot…

Law of cosines

Law of sines

(Draw this)

3.Remember your equations!

c2 = a2 + b2 - 2ab cos C

asinA

=b

sinB=

csinC

A

B

C

c a

b θ = 62

60 N

40 N

170 N

To find the resultant vector ���for this, create a chart and solve for the “x” and “y” components of all three vectors (the 60N, 170N, and 40N)

How do you find the resultant vector for this????

Step 1

3.Remember your equations!

c2 = a2 + b2 - 2ab cos C

asinA

=b

sinB=

csinC

A

B

C

c a

b θ 62 =

60 N

40 N

170 N

Vector 60N 40N 170N total X component Y component

+y

-x

Find the x and y of this vector!

Sin (62) = y / 60 = 53 N

cos (62) = x / 60 = -28 N

Step 2

3.Remember your equations!

c2 = a2 + b2 - 2ab cos C

asinA

=b

sinB=

csinC

A

B

C

c a

b θ 62 =

60 N

40 N

170 N

Vector 60N 40N 170N total X component -28N 0 +170 142N

Y component 53N -40N 0 13N

+y

-x

Find the x and y of this vector!

Sin (62) = y / 60 = 53 N

cos (62) = x / 60 = -28 N

Step 3 Now total the chart

3.Remember your equations!

c2 = a2 + b2 - 2ab cos C

asinA

=b

sinB=

csinC

A

B

C

c a

b

60 N

40 N

Vector 60N 40N 170N total X component -28N 0 +170 142N Y component 53N -40N 0 13N

Step 4 Now Use x and y totals to draw a new diagram!

170N

13N

142N

€

v R

Finally ,calculate the resultant velocity and the direction!!

€

θ

3.Remember your equations!

c2 = a2 + b2 - 2ab cos C

asinA

=b

sinB=

csinC

A

B

C

c a

b

60 N

40 N

Vector 60N 40N 170N total X component -28N 0 +170 142N Y component 53N -40N 0 13N

Step 4 Now Use x and y totals to draw a new diagram!

170N

13N

142N

€

v R

€

θ

C2=(142)2+(13)2= 142.6 N

tan( ) = 13/ 142 = 5.2 NE θSo essentially, a 142.6N force is created by the three forces and is directed at 5.2 degrees North East!

Now find the resultant for this vector

20N

100N

60N

€

θ = 40

I would like a student to put this answer on the board!

The River Problem A boat travels due east across a 100m river at 10 m/s!

+ 10 m/s A

B

100m

1.What is her resultant velocity and path across the river?

2.What is the time required to cross the river?

3.What is her displacement from A to B along the eastern shoreline?

Current is 5m/s

The River Problem A boat travels due east across a 100m river at 10 m/s!

+ 10 m/s A

B

100m

1.What is her velocity and path across the river?

+ 5 m/s

• Move vectors head to tail

• Do pythag’s theorem

• Find direction v

V2=102+52=11.2 m/s

Tan (O) =5/10 =26.5 SE

The River Problem A boat travels due east across a 100m river at 10 m/s!

+ 10 m/s A

B

100m

2.What is the time required to cross the river?

• Stay in the x dimension only

• V = d / t

10m/s = 100m / t t=10 secs

Current is 5m/s

The River Problem A boat travels due east across a 100m river at 10 m/s!

+ 10 m/s A

B

100m

3.What is her displacement from A to B along the eastern shoreline?

•  stay in the y dimension for this

• Dy =vy x time

Dy= 5m/s x 10secs

D=50m

Current is 5m/s

The River Problem A boat travels due east across a 100m river at 10 m/s!

+ 10 m/s

100m

4.Now draw three separate vector triangles for the boat…

•  One for distances

•  One for velocities

•  One for time Current is 5m/s

d v t