Ch 3 Vectors. Vectors What is the difference between a scalar and a vector? A vector is a physical...

Ch 3 Vectors

Vectors

• What is the difference between a scalar and a vector?

• A vector is a physical quantity that has both magnitude and direction

• What are some examples of vectors that we have used in this class?

Vector vs. Scalar

• State whether each of the following quantities is a vector or a scalar:

Position AccelerationVelocity

SpeedDisplacement

Distance

Force

Energy

Temperature

Volume

Vector Vector Scalar

Scalar Scalar

Scalar Scalar

Vector

Vector Vector

Pressure

Scalar

Representing Vectors

• Remember that vectors have magnitude AND direction.

6.8A m��

@32.5

0°

90°

180°

135° 45°

1 2 3

4 5 6

7 8

9 1

0 11 12 13 14 15 16 17 18 19 20

Adding Vectors Graphically

• Pick a scale for your drawing.• Draw the first vector starting at the origin. • Place your protractor at the “Head” of the first

vector to make the correct angle.• Draw the next vector such that it starts at the

head of the first vector. • Continue to line each vector up head to tail.

• Draw the resultant vector.• Measure its length for the magnitude and angle

for its direction.

Resultant Vector

• Resultant Vector is the sum of 2 or more vectors.

• Drawn with a dashed line.2 1.5 @0

mv

s

��

1 0.80 @90m

vs

��

Rv��

Drawn from tail of first vector

to tip of last vector

• Vectors can be added graphically by placing the “Tail” of one vector to the “Head” of the other.

• The Resultant is the sum of components of two or more vectors – The resultant can be found by drawing a vector from

the origin to the head of the last vector

Adding Vectors GraphicallyIf you walked 6 blocks East and then 4 blocks north

What is your displacement?

7.2 @33.6oR Blocks


• When graphically adding vectors:

– The scale must not change– The direction of the reference angle must not

change


• Does the order in which you add the vectors matter?

• 1+2+3=6

• 3+2+1=6

• 2+1+3=6

Adding Vectors Graphically• You walk 5m @ 0o and then turns to walk 6m @90o.

Finally, you turn to walk 8 m at 200°. What is your displacement?

Addition is commutative!

1204@Rd ��


• Multimedia– Vector addition, order does not mater.

Adding Vectors Graphically• Vectors are always

added head to tail• Always measure the

angle from the +x axis.• Vectors are express in

two parts• Given vectors A and B,

find vector C.

4.50@150B

?C

2.5@ 45A

4.5@120oC

Adding Vectors Graphically• Given vectors

A and B, find vector C.

2.5@ 45A C = B + A

4.50@150B

?C

4.5@120oC


• Vector 1:

300.0 m @ 0• Vector 2

450.0 m @ 135 • Vector 3

250.0 m @ 270• What is the Resultant?

• D = A + B + C

70 @105oD m

Relative Velocity

Independence of Vectors

• Multimedia • The river boat• The plane and the wind


A boat travels north at 8m/s across an 80m wide river which flows west at 5m/s . The river is 80m wide


• Perpendicular vector quantities are independent of each other.

• For example in projectile motion– Vx Velocity in the X-direction

– Vy Velocity in the Y-direction

Are independent of each other.

Trig Function Reminders• Trig functions take angles

for input and give ratios for their output.

• Inverse Trig functions take ratios for input and give angles for output.

Trig Functions

Inverse Trig Functions

Adding Force Vectors Analytically

sinOpposite

Hypontenuse

cosadjacent

Hypontenuse

tanopposite

adjacent

hypo

tenus

e

adjacent

opposite

1tan ( )o

a

Components of Vectors

Finding the vector magnitude and direction when you know the components.

Recall: is measured from the positive x axis.

22yx AAA

1tan tany y

x x

A A

A A

Caution: Beware of the tangent function.

Always consider in which quadrant the vector lies when dealing with the tangent function.

III

III IV

8.66

5

1tan (5 / 8.66) 30

-8.66

5

1tan (5 / 8.66)

30

-8.66

-5

1tan ( 5 / 8.66)

30

150

210

8.66

-5

1tan ( 5 / 8.66) 30

330

Adding Vectors Analytically

• Resolve each vector into its horizontal and vertical components

• Add all of the vertical components together

• Add all of the horizontal components together

• Draw a right triangle using the horizontal and vertical resultants


Magnitude Angle X component Y component

4.5N 30o 3.89N 2.25N

7N 210o -6.06N -3.5N

6N 150o -5.19N 3.0N

--------------- ------------ Rx=-7.36N Ry=1.7N

cosxA A 4.5 @30oA N

7 @ 210oB N

6 @150oC N

sinyA A cosxB B cosxC C

sinyB B sinyC C

R=7.55N Angle =-13o+180o = 167o

• Analytically and Graphically add the following vector sets.

• v1 17m/s @ 300• v2 24m/s @ 170• v3 24m/s @ 55o

• v4 19m/s @ 20o

Adding Vectors WS 17

Practice Problem WS6a #117 @300om

sA

24 @170omsB

24 @55omsC

R=22.7m/s Angle = 43.4o

19 @ 20omsD

Magnitude Angle Rx Ry

17 300 8.5 -14.7

24 170 -23.6 4.17

24 55 13.76 19.65

19 20 17.85 6.49

-------------- ------------ 16.51 15.62

Multiplying a Vector by a Scalar

A

A

C = -1/2 A

B = 2A

C

BA

½ A

Adding “-” Vectors

C = A + B

D = A - B

D = A + (- B)

A

B

C -B

D

Add “negative” vectors by keeping the same magnitude but adding 180 degrees to the direction of the original vector.

Vector Concept questions

• What method is used to add vectors graphically?

• How is the resultant vector affected if the force vectors are added in a different order?

• What is equilibrium?


• A vector is to be added graphically, which, if any, of the following may you do the first vector?

a) Rotate it

b) Move it

c) Lengthen it

d) Shorten it


• What is the sum of three vectors that form a triangle?

• If these vectors are forces, what does the imply about the object the forces are acting on?

Adding Vectors

• Graphically and Analytically add the following vector sets.– V1 5.2m/s @ 70– V2 6.4m/s @ 210

– V1 10m/s @ 45– V2 15m/s @ 135

Components of Vectors

• Vector resolution is the process of finding the two component vectors.

Graphical Vector Quiz

On the first part of his flight, Jason flies his plane 5.0 miles due east ( = 5.0 miles @ 0).

He then turns and flies 10.0 miles North West ( 10.0 miles @ 135).

Finally, he turns due south and flies 3.0 miles ( 3.0 miles @ 270). What is his displacement from his takeoff point ?

Quiz Solution

A

BC

Blocks are 1 cm x 1 cm Scale: 1 cm = 2 miles

R

4.5 @117oR mi

• End Ch6 Vectors

Two Body Probems

http://www.clipart.com/en/?nvc_cj=1&AID=10295088&PID=263489



http://www.iclipart.com/search.php?keys=146386&id=104_10_3_17


Protractor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

4.5@120oC


• To find the magnitude of the resultant, measure the length

• To find the direction of the resultant, measure the angle.– The direction is always measured counter

clockwise from the horizontal (east)• Multimedia vector direction


cosxA A 7 @ 45oA N

8 @180oB N

6 @ 270oC N

Ay

Ax

=45o

A=7N

sinyA A

cosxB B

cosxC C

sinyB B

sinyC C

B=8

C=

6

Add the x components togetherAdd the y components togetherCompute the Resultant


Magnitude Angle X Y

7N 45o 4.95N 4.95N

8N 180o -8N 0N

6N 270o 0N -6N

--------------- ------------ Rx=-3.05N Ry=-1.05N

cosxA A 7 @ 45oA N

8 @180oB N

6 @ 270oC N

sinyA A cosxB B cosxC C

sinyB B sinyC C

R=3.22N Angle =19o+180o = 199o

Adding Force Vectors Graphically

7 @ 45oA N

8 @180oB N

6 @ 270oC N

Add the following 3 vectors

3.25 @ 200oR N

Ch 3 Vectors. Vectors What is the difference between a scalar and a vector? A vector is a physical...

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