VectorsChapter 4

Scalar•A quantity with only magnitude

Vector•A quantity with both magnitude

and direction

VectorTail Head

Resultant Vector

•The sum of two or more vectors

Vector Addition•Two addition methods:

•Graphical

•Algebraic

Graphical Vector Addition

•Use the following steps

(1)•Draw any one of the vectors with its tail at the starting point or

origin

(2)•Draw the 2nd vector

with its tail at the head of the first

vector

(3)•Draw the resultant

vector from the starting point of the 1st vector to

the head of the 2nd

(4)•Measure the length of

the resultant to determine the

magnitude of the vector

(5)•Measure the angle to determine the direction

of the vector

Drill:• An insect crawls 4.0 cm

east, then 3.0 cm south. Calculate:

• a) distance traveled

• b) displacement

Practice:• A plane flies 5.0 km west,

then 2500 m south. Calculate:

• a) distance traveled

• b) displacement

Drill:• A bug crawls 3.0 cm west,

then 40.0 mm south. Calculate:

• a) distance traveled

• b) displacement

Drill:• A plane flies 150 m/s east

in a 25 m/s wind blowing towards south. Calculate the plane’s velocity relative to the ground.

Review HW•Problems 5 - 10 on page 71

Adding Vectors with Opposite Signs

•Vector1 + (-Vector2) = Vector1 – Vector2

V1V2

V2 - V1

VR

Practice:• A bird flies 25 m west, then

57 m east. Calculate:

• a) distance traveled

• b) displacement

Practice:• A bird flies 14 m west, then

32 m east, then 21 m west. Calculate:

• a) distance traveled

• b) displacement

A boat travels upstream at 10.0 m/s in a river flowing at 2.5 m/s.

Calculate the velocity of the boat.

Multiple vectors•When adding multiple vectors, just repeat the process of head of first to tail of second etc.

Algebraic

A

BR

Practice:• A car goes 3.0 km west,

then 4.0 km south, then 5.0 km north. Calculate:

• a) distance traveled

• b) displacement

Algebraic

adj

opphyp

Solving the problem

•Sin = opp/hyp

•Cos = adj/hyp

•Tan = opp/adj

Algebraic•R2 = A2 + B2 if right angle

•R2 = A2 + B2 –2ABcos otherwise

A ball rolls 45 m north, then is kicked 60.0 m

west. Calculate the distance & displacement

of the ball.

A ball thrown at 50.0 m/s north from a train moving 50.0 m/s west.

Calculate the velocity of the ball.

A boat travels at 4.0 m/s across in a river flowing at 3.0 m/s. Calculate the

velocity of the boat.

A plane travels at 250 m/s south in a 50.0 m/s wind blowing east to west. Calculate the

velocity of the plane.

A plane travels at 25 m/s south in a 15 m/s wind blowing east to west. Calculate the

velocity of the plane.

Drill: A snail travels at 9.0 cm south then 15.0 cm west then 6.0 cm south. Calculate the displacement of the

snail.

Check HW•Problems 11 – 14

•Page 74

Vector Resolution•Resolving any vector into its x & y components

Vector = 100 units at 37o N o E

y-axis

x-axis

37o

Determine the x & y components

y-axis

Adjacent side37o

Opposite side

Hypotenuse

Solving the problem

•Sin = opp/hyp

•Cos = adj/hyp

•Tan = opp/adj

Solving the problem

•sin = opp/hyp

•opp = hyp x sin

Solving the problem

•cos = adj/hyp

•adj = hyp x cos

Determine the x & y componentsy-axis

Adjacent side = hyp(cos )

Opposite side= hyp(sin )

Hypotenuse = 100 m

Trig Functions• x-component = 100(cos 37o)

= 100(0.80) = 80 units

• y-component = 100(sin 37o)

= 100(0.60) = 60 units

Resolve the following vector into polar or x

& y components:

150 m/s @ 30o N o E

Resolve the following vector into polar or x

& y components:

250 N @ 37o E o S

Resolve the following vector into polar or x

& y components:

7500 N @ 53o

Vector Addition Hint:• When adding multiple

vectors, just add the vector components. Then solve for the final vector.

1) 50 m at 45o E o N2) 45 m at 53o S o W3) 80 m at 30o W o N4) 75 m at 37o N o ECalculate resultant

Equilibrium•When functions applied to any system add up to zero

•Steady State•Homeostasis

Equilibrant•The vector, when added to a set of vectors, would bring the sum of all the vectors back to the zero point or origin.

An automobile is driven 250 km due west, then 150 km

due south. Calculate the resultant vector.

A dog walks 4.0 miles east, then 6.0 miles

north, then 8.0 miles west. Calculate the

resultant vector.

Drill: A cannon fires a projectile at 37o from horizontal at 1250 m/s

Calculate the x & y components.

Check HW: 11 - 14

A jet flies 15 km due west then 25 km

at 53.1o north of west. Calculate the

resultant vector.

1) 9.0 m W2) 800.0 cm S3) 3000.0 mm E4) 0.0035 km NCalculate equilibrant

Resolve a 2.4 kN force vector that is 30.0o from

horizontal into horizontal & vertical

components in N:

1) 2.0 m at 30o

2) 150.0 cm at 37o

3) 3000.0 mm at 53o

4) 0.0040 km at 127o

Calculate equilibrant

The following forces are acting on a point: 1) 5.0 N at 37o

2) 8.0 N at 53o

Calculate equilibrant

A boat travels at 4.0 m/s directly across a river flowing at 3.0 m/s. Calculate the resultant vector.

A boy walks 4.0 miles east, then 6.0

miles north, then 4.0 miles east. Calculate the resultant vector.

A jet flies 15 km due west then 25 km at 53o north of west.

Calculate the resultant vector.

A jet flies 28 km due west then 21 km north. Calculate the

resultant vector.

A human walks 8.0 m due east then 12 m at 30o north of

east. Calculate the resultant vector.

A jet travels 250 miles at 37o north of west.

Resolve the displacement into

north & west components.

1) 50 m at 45o E o N2) 45 m at 53o S o W3) 80 m at 30o W o N4) 75 m at 37o N o ECalculate resultant

A girl walks 25 m due east then 15 m at 37o north of east, the 50.0 m due south. Calculate

the resultant vector.

A girl walks 75 m at 37o north of east, then

75 m at 53o west of north. Calculate the

resultant vector.

1) 50 m at 45o S o W2) 75 m at 53o E o S3) 80 m at 37o N o E4) 75 m at 33o W o N

Calculate resultant

A zombie walks:1) 0.16 km due north2) 90.0 m due east3) 25,000 cm at 37o N o E

Calculate resultant:

A zombie walks:1) 0.30 km at 30o SoW2) 500 m at 45o NoE

Calculate resultant:

A snail crawls:1) 25 cm at 37o WoS2) 400 mm at 30o NoE

Calculate resultant:

A telephone pole has a wire pulling with a 3500 N force attached at 20o

from the top of the pole. Calculate the force

straight down.

A cat walks:1) 90 m due south2) 1600 cm due east3) 5,000 mm at 37o N o E

Calculate resultant:

Forces act on a point:1) 150 N at 53o EoS2) 250 N at 37o SoW3) 0.50 kN at 45o WoS

Calculate resultant:

1) 350 N at 53o WoS2) 150 N at 37o NoW3) 0.25 kN at 45o WoS4) 250 N due E

Calculate resultant:

1) 0.35 kN due west2) 150 N due south3) 0.50 kN at 45o EoN4) 250 N at 37o NoE

Calculate resultant:

1) 0.35 kN due west2) 150 N due south3) 0.50 kN at 45o EoN4) 250 N at 37o NoE

Calculate resultant:

Use graph paper to solve the following:

1) 250 m due east3) 0.50 mm 53o EoN

Calculate resultant:

Solve with trig:1) 0.10 N 37o SoW2) 250 kN 53o EoN3) 150,000 N East

Calculate resultant:

Define the Following:•Distance

•Displacement

•Speed

•Velocity