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Chapter 3

Vectors

Prof. Raymond Lee,revised 9-2-2010

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• Coordinate systems

• Used to describe a point’s position in space

• Coordinate system consists of• fixed reference point called origin

• specific axes with scales & labels

• instructions on how to label a point relative toorigin & axes

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• Cartesian coordinate system

• Also called rectangularcoordinate system

• x- & y- axes intersect atorigin

• Points are labeled (x,y)

(compare Fig. 3-9, p. 41)

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• Polar coordinate system

• Origin & reference line arenoted

• Point is distance r from originin direction of angle ! that’s

CCW from reference line

• Points are labeled (r,!)

(compare Fig. 3-8, p. 41)

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• Polar to Cartesian coordinates

• Based on forming righttriangle from r & !

• x = r cos !

• y = r sin !

(compare figure on p. 43)

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• Cartesian to polar coordinates

• r is hypotenuse & ! an angle

• ! must be CCW from +x axisfor these equations to bevalid

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• Cartesian example

• Cartesian coordinates of apoint in xy plane are (x,y) =(-3.50, -2.50) m, as shown infigure. Find this point’s polarcoordinates.

• Solution: From Eq. 3-6,

•

•

(SJ 2008,p. 54)

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• Vectors & scalars

• A scalar quantity is completely specifiedby a single value with an appropriateunit & has no direction.

• A vector quantity is completelydescribed by a number & appropriateunits + a direction.

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• Vector notation

• When handwritten, use a tilde underscore: A

• When printed, vector will be in boldface: A

• When dealing with just the vector magnitude inprint, use italics: A or |A|

• Vector magnitude has physical units & is always a+ number

~

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• Vector example

• Particle travels from A to Balong path shown by dotted redline, the scalar distance traveled

• Displacement is solid line fromA to B, which is independent ofpath taken between the 2points & is a vector

(compare Fig. 3-2, p. 39)

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• Equality of 2 vectors

• 2 vectors are equal ifthey have samemagnitude & direction

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

• All the vectors shownhere are equal

(compare Fig. 3-1, p. 38)

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• Adding vectors

• When adding vectors, must take into accounttheir directions; units must be the same

• Graphical Methods• Use scale drawings

• Algebraic Methods• More convenient

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• Adding vectors graphically

• Choose a scale

• Draw 1st vector A with appropriate length &in direction specified w.r.t. a coordinatesystem

• Draw 2nd vector with appropriate length &in direction specified w.r.t. a coordinatesystem whose origin is end of vector A & is|| to coordinate system used for A

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• Adding vectors graphically, 2

• Continue drawing thevectors “tip-to-tail”

• Resultant is drawn fromtail of A to tip of lastvector

• Measure length of R &its angle• Use scale factor to

convert R’s length toactual magnitude

(compare Fig. 3-3, p. 39)

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• Adding vectors graphically, 3

• If have many vectors,repeat process until allare included

• Resultant is still drawnfrom origin of 1st vectorto end of last vector

(compare Fig. 3-4, p. 39)

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• Adding vectors, rules

• When we add 2 vectors,sum is independent oforder of addition.• This is commutative law of

addition

• A + B = B + A

(compare Fig. 3-8, p. 39)

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• Adding vectors, rules 2

• When adding ! 3 vectors, sum is independent of theway in which individual vectors are grouped• This is called associative property of addition

• (A + B) + C = A + (B + C)

(Fig. 3-3, p. 39)

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• Adding vectors, rules 3

• When adding vectors, all vectors musthave same units

• All vectors must measure same physicalquantity (e.g., can’t add a displacementto a velocity)

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• Negative of a vector

• Negative of a vector is defined as vectorthat, when added to original vector, !resultant of zero• Represented as –A

• A + (-A) = 0

• – vector has same magnitude, butpoints in opposite direction

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• Subtracting vectors

• Special case of vectoraddition

• If A – B, then use A+(-B)

• Continue with standardvector addition procedure

(compare Fig. 3-6, p. 40)

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• Multiplying or dividing vector by a scalar

• Result of multiplication or division is a vector

• Vector’s magnitude is multiplied or divided byscalar

• If scalar > 0, direction of result is same as oforiginal vector

• If scalar < 0, direction of result is oppositethat of the original vector

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• Vector components

• A component is a part wholevector & is most useful withrectangular components

• Shown are projections ofvector along the x- & y-axes

(compare Fig. 3-8, p. 41)

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• Vector components

• Ax & Ay are component vectors of A

• They are vectors & follow all rules for vectors

• Ax & Ay are scalars & are called thecomponents of A

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• Vector components, 2

• Vector’s x-component is its projectionalong x-axis

• y-component is vector’s projectionalong y-axis

• Then which gives

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• Vector components, 3

• y-component is moved to endof x-component

• Valid since any vector can bemoved || to itself withoutbeing affected• This movement completes

triangle

(compare Fig. 3-8, p. 41)

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• Vector components, 4

• Previous equations are valid only if ! ismeasured with respect to the x-axis

• Components are legs of right triangle whosehypotenuse is A

• May still have to find " w.r.t. + x-axis

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• Vector components, 5

• Components can be +or – & will have sameunits as original vector

• Components’ signsdepend on angle !

(Fig. 3.13, p. 60)

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• Unit vectors

• Unit vector is dimensionless vector withmagnitude = 1.

• Use unit vectors to specify a direction &have no other physical significance

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• Unit vectors, 2

• Symbols

represent unit vectors

• They form a set ofmutually perpendicular(#) vectors

kand,j,i

(compare Fig. 3-13, p. 44)

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• Unit vector notation

• Ax is same as Ax , &Ay is the same as Ay

, etc.

• Write completevector as:

(compare Fig. 3-14,p. 44)

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• Adding vectors using unit vectors

• Using R = A + B

• Then

• & so Rx = Ax + Bx & Ry = Ay + By

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• Trig function warning

• Component equations {Ax = A cos(!) & Ay =A sin(!)} apply only when angle is measured

w.r.t. x-axis (preferably CCW from + x-axis).

• Resultant angle {tan(") = Ay/Ax} gives angle

w.r.t. x-axis.• Think of triangle being formed & corresponding !;

then use appropriate trig functions

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Adding vectors with unit vectors(SJ 2008 Fig. 3.16, p. 61)

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• Adding vectors using 3D unit vectors

• Using R = A + B

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

• etc.

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• Example: Taking a hike

• A hiker starts a trip by first walking 25.0 kmSE from her car. She stops & sets up her tentfor the night. On 2nd day, she walks 40.0 kmin a direction 60.0° N of E, at which point shediscovers a forest ranger’s tower.

{SJ 2008, p. 63}

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• hiking example, p. 2

• (A) Determine components ofhiker’s displacement for each day.

Solution: We conceptualize problem by drawing asketch as in figure above. If we denote displacementvectors on 1st & 2nd days by A & B respectively, & usecar as coordinate origin, we get vectors shown in figure.Drawing resultant R, we can now categorize this problemas an addition of 2 vectors.

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• hiking example, p. 3

• Analyze this problem using vectorcomponents. Displacement A hasmagnitude = 25.0 km & is directed45.0° below +x axis.

From Eq. 3-5 (p. 41), its components are:

– value of Ay indicates that hiker walks in –y direction on 1stday. Signs of Ax & Ay also are evident from figure above.

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• hiking example, p. 4

• 2nd displacement B has amagnitude = 40.0 km & is60.0° N of E.

Its components are:

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• hiking example, p. 5

• (B) Determine the components ofthe hiker’s resultant displacementR for the trip. Find an expressionfor R in terms of unit vectors.

Solution: The resultant displacement for the trip R = A + Bhas components given by Eqs. 3-10 & 3-11 (p. 44):

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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• hiking example, p. 6

• Using Eq. 3-6 (p. 42), we find thatvector R has a magnitude = 41.3km & points 24.1° N of E.

Now finalize. Units of R are km, which is reasonable for adisplacement. From graphical representation in figure, estimatethat hiker’s final position ~ (38 km, 17 km), consistent withcomponents of R in final result. Also, both components of R > 0,putting final position in 1st quadrant of coordinate system, alsoconsistent with figure.

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• Problem-solving strategy: Adding vectors

• Select a coordinate system

• Try to select a system that minimizes # ofcomponents you must deal with

• Draw a sketch of vectors & label each

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• Problem-solving strategy: Adding vectors, 2

• Find x & y components of each vector & x & ycomponents of resultant vector

• Find z components if necessary

• Use Pythagorean theorem to get resultant’smagnitude & tangent function to get itsdirection

• Other appropriate trig functions may be needed