C2 – Vectors C3 – Interactions transfer...

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General Physics GP7-Vectors (Ch 4) 1

• C2 – Vectors• C3 – Interactions transfer momentum

Solutions to HW

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Vectors and Scalars

• scalar quantity• completely specified by a single value

with an appropriate unit and has nodirection.

• vector quantity• completely described by a number and

appropriate units plus a direction.

Vectors and Scalars

• scalar quantity• completely specified by a single value with an

appropriate unit and has no direction.• Use an italic letter:

• vector quantity• completely described by a number and

appropriate units plus a direction.• Use an arrow:

Vector Example

• A particle travels from Ato B along the pathshown by the dottedred line• 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 equal ifthey have the samemagnitude and the samedirection

• if A = B and theypoint along parallel lines

• All of the vectors shown inthe diagram at right areequal

Adding Vectors Graphically

• Draw the vectors “tip-to-tail”

• The resultant is drawnfrom the origin ofto the end of the lastvector

• Measure the length ofand its angle• Use the scale factor to

convert length to actualmagnitude

Adding Multiple Vectors

• tip-to-tail for allvectors

• The resultant, ,is still drawn fromthe origin of the firstvector to the end ofthe last vector

Adding Vectors, Rules

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

commutative lawof addition

rrrr+=+

Adding Vectors, Rules cont.

• When adding three ormore vectors, their sumis independent of theway in which theindividual vectors aregrouped• This is called the

Associative Propertyof Addition

( ) ( )CBACBA

rrrrrr++=++

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 as thevector that, when added to the originalvector, gives a resultant of zero• Represented as –

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

( ) 0=!+ AA

Subtracting Vectors

• Special case ofvector addition

• If , then use

• Continue withstandard vectoraddition procedure

Vector representation

• A vector has both magnitude (numbervalue) and direction.

zpypxppzyxˆˆˆ ++=

Show magnitude or value Show direction

Column Vector

• How can we describe this vector using ausing column vector?

zpypxppzyxˆˆˆ ++=

rtypical vector notation

column vector notation

Components of a Vector

• A component is a part• We will use rectangular

components• These are the projections of the

vector along the x- and y-axes

• are thecomponent vectors of• They are vectors and follow all

the rules for vectors

• Ax and Ay are scalars, and willbe referred to as thecomponents of

yx AArr

Magnitude and Direction

• A vector may also be represented by it’smagnitude and direction.

• Magnitude

• The magnitude of the vector hasphysical units

• The magnitude of a vector is always apositive number

( ) 222

zyxppppmagp ++==

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

Think-Pair-Share

• A boat is crossing a river that flowsfrom North to South at a rate of 3 m/s.The boat starts at the east end of theriver and heads directly west with aspeed of 4 m/s.(a) What is the boats total velocity?(b) If the river is 1000 m in the E-Wdirection, how long does it take theboat to cross?

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,

r A = Ax

ˆ x + Ayˆ y

Components of a Vector, 3

• The previous equations arevalid only if θ is measuredwith respect to the x-axis

• The components are the legsof the right triangle whosehypotenuse is A

Components of a Vector, final

• The components canbe positive ornegative and willhave the same unitsas the originalvector

• The signs of thecomponents willdepend on the angle

Think-Pair-Share

• A displacement vector in the x-y plane is15 m long and directed at an angle of 30degrees above the positive x axis.Determine(a) the x-component(b) the y-component

Think-Pair-Share

• A displacement vector in the x-y plane is10 m long and directed at an angle of 190degrees counterclockwise from the positivex axis. Determine(a) the x-component(b) the y-component

Think-Pair-Share

• 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?

Think-Pair-Share

• 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?

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

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

• Solution:

Unit Vectors

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

• Unit vectors are used to specify a direction(x,y, and z) and have no other physicalsignificance

• The symbols for x, y, z are

• They form a set of mutually perpendicularvectors

zyx ˆ and ˆ,ˆ

Unit Vectors in Vector Notation

• The complete vectorcan be expressed as

zAyAxAA zyxxˆˆˆ ++=

Adding Vectors Using UnitVectors

• Using• Then

• and so Rx = Ax + Bx

• and Ry = Ay + By

( ) ( )

yBAxBAR

yBxByAxAR

ˆˆˆˆ

Ax + Bx

Ay + By

Adding Vectors with Unit Vectors

Adding Vectors Using UnitVectors – Three Directions

• Using

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

( ) ( )

( ) ( ) ( )

+++++=

zzyyxx

zyxzyx

zRyRxRR

zBAyBAxBAR

zByBxBzAyAxAR

ˆˆˆ

ˆˆˆˆˆˆ

Think-Pair-Share

• (a) In unit-vector notation, what is the sumof , where

a = (4.0 m) i + (3.0 m) j

b = (-13.0 m) i + (7.0 m) j

• What are the (b) magnitude and (c) directionof (relative to )?

Think-Pair-Share

• Repeat the problem using columnvector notation.

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)

Polar to Cartesian Coordinates

• Based onforming a righttriangle from rand θ

• x = r cos θ• y = r sin θ• SOH-CAH-TOA

Cartesian to Polar Coordinates

• r is the hypotenuse andθ an angle

• θ must be counter-clockwise frompositive x axis forthese equations to bevalid

C2 – Vectors C3 – Interactions transfer...

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