Chapter 1 – Math Chapter 1 – Math ReviewReview

Surveyors use accurate measures of magnitudes and

directions to create scaled maps of large regions.

VectorsVectors

Objectives: After completing Objectives: After completing this module, you should be this module, you should be able to:able to:• Demonstrate that you meet mathematics Demonstrate that you meet mathematics

expectations: unit analysis, algebra, scientific expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry.notation, and right-triangle trigonometry.

• Define and give examples of scalar and Define and give examples of scalar and vector quantities.vector quantities.

• Determine the components of a given vector.Determine the components of a given vector.

• Find the resultant of two or more vectors.Find the resultant of two or more vectors.

• Demonstrate that you meet mathematics Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry.notation, and right-triangle trigonometry.

• Define and give examples of scalar and Define and give examples of scalar and vector quantities.vector quantities.

• Determine the components of a given vector.Determine the components of a given vector.

• Find the resultant of two or more vectors.Find the resultant of two or more vectors.

ExpectationsExpectations

• You must be able convert units of You must be able convert units of measure for physical quantities.measure for physical quantities.

Convert 40 m/s into kilometers per hour.

40--- x ---------- x -------- = 144 km/h

m

s

1 km

1000 m

3600 s

1 h

Expectations (Continued)Expectations (Continued)

• You must be able to work in You must be able to work in scientific notation.scientific notation.

Evaluate the following:

(6.67 x 10-11)(4 x 10-3)(2)

(8.77 x 10-3)2 F = -------- = ------------

Gmm’

r2

F = 6.94 x 10-9 N = 6.94 nNF = 6.94 x 10-9 N = 6.94 nN

Expectations (Continued)Expectations (Continued)

• You must be familiar with SI prefixesYou must be familiar with SI prefixes

The meter (m) 1 m = 1 x 100 m

1 Gm = 1 x 109 m 1 nm = 1 x 10-9 m

1 Mm = 1 x 106 m 1 m = 1 x 10-

6 m

1 km = 1 x 103 m 1 mm = 1 x 10-

3 m

Expectations (Continued)Expectations (Continued)

• You must have mastered right-You must have mastered right-triangle trigonometry. triangle trigonometry.

y

x

R

y = R sin y = R sin

x = R cos x = R cos

siny

R

cosx

R

tany

x R2 = x2 +

y2

R2 = x2 + y2

Science of MeasurementScience of Measurement

We begin with the measurement of length: its magnitude and its direction.

We begin with the measurement of length: its magnitude and its direction.

LengtLengthh

WeighWeightt

TimeTime

Some Physics QuantitiesSome Physics Quantities

Vector - quantity with both magnitude (size) and directionVector - quantity with both magnitude (size) and direction

Scalar - quantity with magnitude onlyScalar - quantity with magnitude only

VectorsVectors::

•

DisplacementDisplacement

• VelocityVelocity

• AccelerationAcceleration

• MomentumMomentum

• ForceForce

Scalars:Scalars:

• DistanceDistance

• SpeedSpeed

• TimeTime

• MassMass

• EnergyEnergy

Mass vs. WeightMass vs. Weight

On the moon, your mass would be the same, but On the moon, your mass would be the same, but the magnitude of your weight would be less.the magnitude of your weight would be less.

MassMass

• Scalar (no direction)Scalar (no direction)

• Measures the amount of matter in an objectMeasures the amount of matter in an object

WeightWeight

• Vector (points toward center of Earth)Vector (points toward center of Earth)

• Force of gravity on an objectForce of gravity on an object

VectorsVectors

• The length of the The length of the arrow represents arrow represents the magnitude the magnitude (how far, how fast, (how far, how fast, how strong, etc, how strong, etc, depending on the depending on the type of vector).type of vector).

• The arrow points in The arrow points in the directions of the directions of the force, motion, the force, motion, displacement, etc. displacement, etc. It is often specified It is often specified by an angle.by an angle.

Vectors are represented with arrowsVectors are represented with arrows

42°42°

5 5 m/sm/s

UnitsUnits

Quantity . . . Unit (symbol) Quantity . . . Unit (symbol)

• Displacement & Distance . . . meter (m)Displacement & Distance . . . meter (m)

• Time . . . second (s)Time . . . second (s)

• Velocity & Speed . . . (m/s)Velocity & Speed . . . (m/s)

• Acceleration . . . (m/sAcceleration . . . (m/s22))

• Mass . . . kilogram (kg)Mass . . . kilogram (kg)

• Momentum . . . (kg · m/s)Momentum . . . (kg · m/s)

• Force . . .Newton (N)Force . . .Newton (N)

• Energy . . . Joule (J)Energy . . . Joule (J)

Units are not the same as quantities!Units are not the same as quantities!

SI PrefixesSI Prefixes

pico p 10-12

nano n 10-9

micro µ 10-6

milli m 10-3

centi c 10-2

kilo k 103

mega M 106

giga G 109

tera T 1012

Little GuysLittle Guys Big GuysBig Guys

Distance: A Scalar Distance: A Scalar QuantityQuantity

A scalar quantity:

Contains magnitude only and consists of a number and a unit.

(20 m, 40 mi/h, 10 gal)

A

B

DistanceDistance is the length of the actual is the length of the actual path taken by an object.path taken by an object.

DistanceDistance is the length of the actual is the length of the actual path taken by an object.path taken by an object.

s = 20 m

Displacement—A Vector Displacement—A Vector QuantityQuantity

A vector quantity:

Contains magnitude AND direction, a number, unit & angle.

(12 m, 300; 8 km/h, N)

A

BD = 12 m, 20o

• DisplacementDisplacement is the straight-line is the straight-line separation of two points in a separation of two points in a specified direction.specified direction.

• DisplacementDisplacement is the straight-line is the straight-line separation of two points in a separation of two points in a specified direction.specified direction.

Distance and Distance and DisplacementDisplacement

Net Net displacement:displacement:4 m,E4 m,E

6 6 m,Wm,W

D

What is the What is the distance traveled?distance traveled?

10 m !!

DD = 2 m, W= 2 m, W

• DisplacementDisplacement is the is the x x or or yy coordinate of position. Consider a coordinate of position. Consider a car that travels 4 m, E then 6 m, car that travels 4 m, E then 6 m, W.W.

• DisplacementDisplacement is the is the x x or or yy coordinate of position. Consider a coordinate of position. Consider a car that travels 4 m, E then 6 m, car that travels 4 m, E then 6 m, W.W.

xx = +4= +4xx = -2= -2

Identifying DirectionIdentifying Direction

A common way of identifying direction A common way of identifying direction is by reference to East, North, West, is by reference to East, North, West, and South. (Locate points below.)and South. (Locate points below.)

A common way of identifying direction A common way of identifying direction is by reference to East, North, West, is by reference to East, North, West, and South. (Locate points below.)and South. (Locate points below.)

40 m, 5040 m, 50oo N of E N of E

EW

S

N

40 m, 60o N of W40 m, 60o W of S40 m, 60o S of E

Length = 40 m

5050oo60o

60o60o

Identifying DirectionIdentifying Direction

Write the angles shown below by using Write the angles shown below by using references to east, south, west, north.references to east, south, west, north.Write the angles shown below by using Write the angles shown below by using references to east, south, west, north.references to east, south, west, north.

EW

S

N45o

EW

N

50o

S

Click to see the Answers . . .Click to see the Answers . . .500 S of E500 S of E

450 W of N450 W of N

Vectors and Polar Vectors and Polar CoordinatesCoordinates

Polar coordinates (Polar coordinates (R,R,) are an ) are an excellent way to express vectors. excellent way to express vectors. Consider the vector Consider the vector 40 m, 5040 m, 500 0 N of EN of E,, for example.for example.

Polar coordinates (Polar coordinates (R,R,) are an ) are an excellent way to express vectors. excellent way to express vectors. Consider the vector Consider the vector 40 m, 5040 m, 500 0 N of EN of E,, for example.for example.

0o

180o

270o

90o

0o

180o

270o

90o

RR

RR is the is the magnitudemagnitude and and is the is the directiondirection..

40 40 mm5050oo

Vectors and Polar Vectors and Polar CoordinatesCoordinates

(R,(R,) = 40 m, 50) = 40 m, 50oo

(R,(R,) = 40 m, ) = 40 m, 120120oo (R,(R,) = 40 m, 210) = 40 m, 210oo

(R,(R,) = 40 m, ) = 40 m, 300300oo

5050oo60o

60o60o

0o180o

270o

90o

120o

Polar coordinates (Polar coordinates (R,R,) are given for ) are given for each of four possible quadrants:each of four possible quadrants:Polar coordinates (Polar coordinates (R,R,) are given for ) are given for each of four possible quadrants:each of four possible quadrants:

210o

3000

Example 1:Example 1: Find the height of a Find the height of a building if it casts a shadow building if it casts a shadow 90 m90 m long and the indicated angle is long and the indicated angle is 3030oo..

90 m

300

The height h is opposite 300

and the known adjacent side is 90 m.

h

h = (90 m) tan 30o

h = 57.7 mh = 57.7 m

0tan 3090 m

opp h

adj

Finding Components of Finding Components of VectorsVectorsA component is the effect of a vector along other directions. The x and y components of the vector (R, are illustrated below.

x

yR

x = R cos

y = R sin

Finding components:

Polar to Rectangular Conversions

Example 2:Example 2: A person walks A person walks 400 m400 m in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

x

yR

x = ?

y = ?400 m

E

N

The y-component (N) is OPP:

The x-component (E) is ADJ:

x = R cos y = R sin

E

N

Example 2 (Cont.):Example 2 (Cont.): A A 400-m400-m walk walk in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

x = R cos

x = (400 m) cos 30o

= +346 m, E

x = ?

y = ?400 m

E

N Note:Note: xx is the side is the side adjacentadjacent to angle to angle

303000

ADJADJ = HYP x = HYP x CosCos 303000

The x-component The x-component is:is:RRxx = = +346 m+346 m

Example 2 (Cont.):Example 2 (Cont.): A A 400-m400-m walk walk in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

y = R sin

y = (400 m) sin 30o

= + 200 m, N

x = ?

y = ?400 m

E

N

OPPOPP = HYP x = HYP x SinSin 303000

The y-component The y-component is:is:RRyy = = +200 m+200 m

Note:Note: yy is the side is the side oppositeopposite to angle to angle

303000

Example 2 (Cont.):Example 2 (Cont.): A A 400-m400-m walk walk in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

Rx = +346 m

Ry = +200 m

400 m

E

NThe x- and y- The x- and y- components components are are eacheach + in + in

the first the first quadrantquadrant

Solution: The person is displaced 346 m east and 200 m north of the original

position.

Signs for Rectangular Signs for Rectangular CoordinatesCoordinates

First Quadrant:

R is positive (+)

0o > < 90o

x = +; y = +x = R cos y = R sin

+

+

0o

90o

R

Signs for Rectangular Signs for Rectangular CoordinatesCoordinates

Second Quadrant:

R is positive (+)

90o > < 180o

x = - ; y = +x = R cos y = R sin

+R

180o

90o

Signs for Rectangular Signs for Rectangular CoordinatesCoordinates

Third Quadrant:

R is positive (+)

180o > < 270o

x = - y = - x = R cos y = R sin

-R

180o

270o

Signs for Rectangular Signs for Rectangular CoordinatesCoordinates

Fourth Quadrant:

R is positive (+)

270o > < 360o

x = + y = -

x = R cos y = R sin

360o+

R

270o

Resultant of Perpendicular Resultant of Perpendicular VectorsVectorsFinding resultant of two perpendicular vectors is like changing from rectangular to polar coord.

R is always positive; is from + x axis

2 2R x y 2 2R x y

tany

x tan

y

x x

yR