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Chapter 3

Vectors

Coordinate Systems

• Used to describe the position of a pointin space

• Coordinate system consists of• a fixed reference point called the origin• specific axes with scales and labels• instructions on how to label a point relative

to the origin and the axes

Cartesian Coordinate System

• Also calledrectangularcoordinate system

• x- and y- axesintersect at theorigin

• Points are labeled(x,y)

Polar Coordinate System

• Origin and referenceline are noted

• Point is distance rfrom the origin in thedirection of angle θ,counter-clockwisefrom reference line

• Points are labeled (r,θ)

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Polar to Cartesian Coordinates

• Based onforming a righttriangle from rand θ

• x = r cos θ• y = r sin θ

• SOH-CAH-TOA

H

O

A

Cartesian to Polar Coordinates

• r is the hypotenuse andθ an angle

• θ must be counter-clockwise from positivex axis for theseequations to be valid

Think-Pair-Share

• Jane leaves her houseand walks 5.0 blockseast and then proceedsnorth until she is 7.8blocks from home at anangle of 50 degreesNorth of east. Howmany blocks north didshe travel?

Example 3.1

• The Cartesian coordinates of apoint in the xy plane are (x,y) =(-3.50, -2.50) m, as shown inthe figure. Find the polarcoordinates of this point.

• Solution: From Equation 3.4,

and from Equation 3.3,

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Vectors and Scalars

• scalar quantity• completely specified by a single value with

an appropriate unit and has no direction.

• vector quantity• completely described by a number and

appropriate units plus a direction.

Vector Notation

• When handwritten, use an arrow: A• When printed, will be in bold print: A• When dealing with just the magnitude of a

vector in print, an italic letter will be used:• A or |A|

• The magnitude of the vector has physicalunits

• The magnitude of a vector is always apositive number

Vector Example

• A particle travels from Ato B along the pathshown by the dotted redline• This is the distance

traveled and is a scalar

• The displacement isthe solid line from A toB• The displacement is

independent of the pathtaken between the twopoints

• Displacement is a vector

Equality of Two Vectors

• Two vectors are equalif they have the samemagnitude and thesame direction

• A = B if A = B and theypoint along parallel lines

• All of the vectors shownare equal

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Adding Vectors Graphically

• Draw the vectors “tip-to-tail”

• The resultant is drawnfrom the origin of A tothe end of the lastvector

• Measure the length of Rand its angle• Use the scale factor to

convert length to actualmagnitude

Adding Multiple Vectors

• tip-to-tail for all vectors

• The resultant, R, isstill drawn from theorigin of the first vectorto the end of the lastvector

Adding Vectors, Rules

• When two vectorsare added, the sumis independent ofthe order of theaddition.• This is the

commutative law ofaddition

• A + B = B + A

Adding Vectors, Rules cont.

• When adding three or more vectors, their sum isindependent of the way in which the individualvectors are grouped• This is called the Associative Property of Addition• (A + B) + C = A + (B + C)

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Adding Vectors, Rules final

• When adding vectors, all of the vectorsmust have the same units

• All of the vectors must be of the sametype of quantity• For example, you cannot add a

displacement to a velocity

Negative of a Vector

• The negative of a vector is defined asthe vector that, when added to theoriginal vector, gives a resultant of zero• Represented as –A• A + (-A) = 0

• The negative of the vector will have thesame magnitude, but point in theopposite direction

Subtracting Vectors

• Special case ofvector addition

• If A – B, then useA+(-B)

• Continue withstandard vectoraddition procedure

Multiplying or Dividing a Vectorby a Scalar

• The result of the multiplication or division is avector

• The magnitude of the vector is multiplied ordivided by the scalar

• If the scalar is positive, the direction of theresult is the same as of the original vector

• If the scalar is negative, the direction of theresult is opposite that of the original vector

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Components of a Vector

• A component is apart

• It is useful to userectangularcomponents• These are the

projections of thevector along the x-and y-axes

Vector Component Terminology

• Ax and Ay are the component vectorsof A• They are vectors and follow all the rules for

vectors

• Ax and Ay are scalars, and will bereferred to as the components of A

Components of a Vector, 2

• The x-component of a vector is the projectionalong the x-axis

• The y-component of a vector is the projectionalong the y-axis

• Then,

Components of a Vector, 3

• The y-component ismoved to the end of thex-component

• This is due to the factthat any vector can bemoved parallel to itselfwithout being affected• This completes the

triangle

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Components of a Vector, 4

• The previous equations are valid only if θ ismeasured with respect to the x-axis

• The components are the legs of the righttriangle whose hypotenuse is A

• May still have to find θ with respect to the positivex-axis

Components of a Vector, final

• The componentscan be positive ornegative and willhave the same unitsas the original vector

• The signs of thecomponents willdepend on the angle

Ch 3: Problem 3

• A fly lands on one wall of a room. Thelower left-hand corner of the wall isselected as the origin of the two-dimensional Cartesian coordinatesystem. If the fly is located at the pointhaving coordinates (2.00, 1.00) m, (a)how far is it from the corner of theroom? (b) What is its location in polarcoordinates?

Ch 3: Problem 20

• A person walks 25.00 north of east for3.10 km. How far would she have towalk due north and due east to arrive atthe same location?

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Unit Vectors

• A unit vector is a dimensionless vectorwith a magnitude of exactly 1.

• Unit vectors are used to specify adirection and have no other physicalsignificance

Unit Vectors, cont.

• The symbols

represent unit vectors• They form a set of

mutually perpendicularvectors

kand,j,i

Unit Vectors in Vector Notation

• Ax = Ax

• Ay = Ay

• The complete vectorcan be expressed as

Adding Vectors Using UnitVectors

• Using R = A + B

• Then

• and so Rx = Ax + Bx and Ry = Ay + By

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Trig Function Warning

• The component equations (Ax = A cos θ andAy = A sin θ) apply only when the angle ismeasured with respect to the x-axis(preferably counter-clockwise from thepositive x-axis).

• The resultant angle (tan θ = Ay / Ax) gives theangle with respect to the x-axis.• You can always think about the actual triangle

being formed and what angle you know and applythe appropriate trig functions

Adding Vectors with Unit Vectors

Adding Vectors Using UnitVectors – Three Directions

• Using R = A + B

• Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz

Example 3.5: Taking a Hike

• A hiker begins a trip by first walking 25.0 kmsoutheast from her car. She stops and setsup her tent for the night. On the second day,she walks 40.0 km in a direction 60.0° northof east, at which point she discovers a forestranger’s tower.

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Example 3.5

• (A) Determine the componentsof the hiker’s displacement foreach day.

Solution: We conceptualize the problem by drawing asketch as in the figure above.

A = first day

B = second day

Drawing the resultant R, we can now categorize thisproblem as an addition of two vectors.

Example 3.5

• Analyze using vectorcomponents.

• Displacement A• magnitude of 25.0 km• 45.0° below the positive x axis.

From Equations 3.8 and 3.9, its components are:

The negative value of Ay indicates that the hiker walks in thenegative y direction on the first day. The signs of Ax and Ayalso are evident from the figure above.

Example 3.5

• The second displacement Bhas a magnitude of 40.0 kmand is 60.0° north of east.

Its components are:

Example 3.5

• (B) Determine the componentsof the hiker’s resultantdisplacement R for the trip. Findan expression for R in terms ofunit vectors.

Solution: The resultant displacement for the trip R = A + Bhas components given by Equation 3.15:

Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km

Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km

In unit-vector form, we can write the total displacement as

R = (37.7 + 16.9 ) kmji

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Example 3.5

• Using Equations 3.16 and 3.17,we find that the vector R has amagnitude of 41.3 km and isdirected 24.1° north of east.