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Vectors. Chapter 4. Scalar. A quantity with only magnitude. Vector. A quantity with both magnitude and direction. Vector. Tail Head. Resultant Vector. The sum of two or more vectors. Vector Addition. Two addition methods: Graphical Algebraic. Graphical Vector Addition.

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Vectors

Vectors

Chapter 4


Scalar
Scalar

  • A quantity with only magnitude


Vector

  • A quantity with both magnitude and direction


Vector

Tail 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
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
Review HW

  • Problems 5 - 10 on page 71


Adding vectors with opposite signs
Adding Vectors with Opposite Signs

  • Vector1 + (-Vector2) = Vector1 – Vector2


V2

V1

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
Multiple vectors 2.5 m/s. Calculate the velocity of the boat.

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


Algebraic 2.5 m/s. Calculate the velocity of the boat.

R

B

q

A


Practice: 2.5 m/s. Calculate the velocity of the boat.

  • A car goes 3.0 km west, then 4.0 km south, then 5.0 km north. Calculate:

  • a) distance traveled

  • b) displacement


Algebraic 2.5 m/s. Calculate the velocity of the boat.

hyp

opp

q

adj


Solving the problem 2.5 m/s. Calculate the velocity of the boat.

  • Sin q = opp/hyp

  • Cos q = adj/hyp

  • Tan q = opp/adj


Algebraic 2.5 m/s. Calculate the velocity of the boat.

  • R2 = A2 + B2

  • if right angle

  • R2 = A2 + B2 –2ABcos q otherwise


A ball rolls 45 m north then is kicked 60 0 m west calculate the distance displacement of the ball
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
Check HW then 6.0 cm south. Calculate the displacement of the snail.

  • Problems 11 – 14

  • Page 74


Vector resolution
Vector Resolution then 6.0 cm south. Calculate the displacement of the snail.

  • Resolving any vector into its x & y components


y-axis then 6.0 cm south. Calculate the displacement of the snail.

Vector = 100 units at 37o N o E

37o

x-axis


y-axis then 6.0 cm south. Calculate the displacement of the snail.

Determine the x & y components

Hypotenuse

Opposite side

37o

Adjacent side


Solving the problem then 6.0 cm south. Calculate the displacement of the snail.

  • Sin q = opp/hyp

  • Cos q = adj/hyp

  • Tan q = opp/adj


Solving the problem then 6.0 cm south. Calculate the displacement of the snail.

  • sin q = opp/hyp

  • opp = hyp x sin q


Solving the problem then 6.0 cm south. Calculate the displacement of the snail.

  • cos q = adj/hyp

  • adj = hyp x cos q


y-axis then 6.0 cm south. Calculate the displacement of the snail.

Determine the x & y components

Hypotenuse = 100 m

Opposite side

= hyp(sin q)

q = 37o

Adjacent side = hyp(cos q)


Trig functions
Trig Functions then 6.0 cm south. Calculate the displacement of the snail.

  • 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
Resolve the following vector into polar or x & y components: then 6.0 cm south. Calculate the displacement of the snail.

150 m/s @ 30o N o E


Resolve the following vector into polar or x & y components: then 6.0 cm south. Calculate the displacement of the snail.

250 N @ 37o E o S


Resolve the following vector into polar or x & y components: then 6.0 cm south. Calculate the displacement of the snail.

7500 N @ 53o


Vector addition hint
Vector Addition Hint: then 6.0 cm south. Calculate the displacement of the snail.

  • When adding multiple vectors, just add the vector components. Then solve for the final vector.


  • 50 m at 45 then 6.0 cm south. Calculate the displacement of the snail.o E o N

  • 2) 45 m at 53o S o W

  • 3) 80 m at 30o W o N

  • 4) 75 m at 37o N o E

  • Calculate resultant


Equilibrium
Equilibrium then 6.0 cm south. Calculate the displacement of the snail.

  • When functions applied to any system add up to zero

  • Steady State

  • Homeostasis


Equilibrant then 6.0 cm south. Calculate the displacement of the snail.

  • 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 37 miles west. Calculate the resultant vector.o from horizontal at 1250 m/s Calculate the x & y components.


Check HW: 11 - 14 miles west. Calculate the resultant vector.


A jet flies 15 km due west then 25 km at 53.1 miles west. Calculate the resultant vector.o north of west. Calculate the resultant vector.


  • 9.0 m W miles west. Calculate the resultant vector.

  • 2) 800.0 cm S

  • 3) 3000.0 mm E

  • 4) 0.0035 km N

  • Calculate equilibrant


Resolve a 2.4 kN force vector that is 30.0 miles west. Calculate the resultant vector.o from horizontal into horizontal & vertical components in N:


  • 2.0 m at 30 miles west. Calculate the resultant vector.o

  • 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 37 miles west. Calculate the resultant vector.o

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 53 miles east. Calculate the resultant vector.o north of west. Calculate the resultant vector.



A dog walks 8.0 m due east then 15 m at 37 resultant vector.o north of east. Calculate the resultant vector.


A jet travels 250 miles at 37 resultant vector.o north of west. Resolve the displacement into north & west components.


  • 50 m at 45 resultant vector.o E o N

  • 2) 45 m at 53o S o W

  • 3) 80 m at 30o W o N

  • 4) 75 m at 37o N o E

  • Calculate resultant


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


A girl walks 75 m at 37 resultant vector.o north of east, then 75 m at 53o west of north. Calculate the resultant vector.


  • 50 m at 45 resultant vector.o S o W

  • 2) 75 m at 53o E o S

  • 3) 80 m at 37o N o E

  • 4) 75 m at 33o W o N

  • Calculate resultant


Drill: A dog walks: resultant vector.

1) 0.16 km due north

2) 90.0 m due east

3) 25,000 cm at 37o N o E

Calculate: Res. & Eq.


Check hw1
Check HW resultant vector.

  • Problems 31 & 31

  • Page 79


A zombie walks: resultant vector.

1) 0.30 km at 30oSoW

2) 500 m at 45o NoE

Calculate resultant:


Drill: A snail crawls: resultant vector.

1) 25 cm at 37oWoS

2) 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: attached at 20

1) 9.0 m due south

2) 1500 cm due east

3) 5,000 mm at 37o N o E

Calculate resultant:


Forces act on a point: attached at 20

1) 150 N at 53o EoS

2) 250 N at 37o SoW

3) 0.50 kN at 45oWoS

Calculate resultant:


1) 350 N at 53 attached at 20o WoS

2) 150 N at 37o NoW

3) 0.25 kN at 45oWoS

4) 250 N due E

Calculate resultant:


1) 0.35 kN due west attached at 20

2) 150 N due south

3) 0.50 kN at 45oEoN

4) 250 N at 37o NoE

Calculate resultant:


Use graph paper to solve the following: attached at 20

1) 250 mm due east

3) 0.50 mm 53oEoN

Calculate resultant:


Drill & Collect HW: Solve the following: attached at 20

1) 360 m due west

3) 0.27 km due north

Calculate resultant:


HW: Solve with trig: attached at 20

1) 0.10 MN 37o SoW

2) 250 kN 53oEoN

3) 150,000 N East

Calculate resultant:


Use graph paper to solve the following: attached at 20

1) 3.0 m due west

3) 15 m 53oEoN

Calculate resultant:


1) 0.35 km due west attached at 20

2) 250 m due south

3) 0.50 km at 45oEoN

4) 150 m at 37o NoE

Calculate resultant:


Define the Following: attached at 20

  • Scalar

  • Vector

  • Magnitude

  • Direction


Define the Following: attached at 20

  • Distance

  • Displacement

  • Speed

  • Velocity


Test review
Test Review attached at 20


Terms to Define: attached at 20

  • Equilibrant

  • Vector Resultant

  • Scalar

  • Vector

  • Vector Resolution


Metric Prefixes: attached at 20

  • Centi Kilo

  • Giga Mega

  • Micro Milli

  • Nano


Trig Functions: attached at 20

  • Sin q Pytha-

  • Cos q Theorem

  • Tan q

  • Law of Cosines


Add the 3 Vectors Graphically: attached at 20

  • 50.0 m west

  • 90.0 m north

  • 170 m east


Add the 2 Vectors Mathematically: attached at 20

  • 20.0 m west

  • 0.10 km @ 37oNoE


Resolve the Vector into x & y comp: attached at 20

  • 0.450 km @ 53o SoW


Add the 3 Vectors using vector components: attached at 20

  • 75 m @ 37o NoW

  • 90.0 m @ 37o NoE

  • 150 m @ 53o SoW


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