ScalarsScalars

A scalar is any physical quantity that can be A scalar is any physical quantity that can be completely characterized by its magnitude (by a completely characterized by its magnitude (by a number value)number value)

Mathematical operations involving scalars – follow Mathematical operations involving scalars – follow rules of elementary algebrarules of elementary algebra

In text represented with letters in In text represented with letters in italicitalic type type Examples: mass, volume, lengthExamples: mass, volume, length

VectorsVectors

Possess a magnitude, direction, and sensePossess a magnitude, direction, and sense Represented graphically by an arrow (tail, tip or head) Represented graphically by an arrow (tail, tip or head)

SHOWSHOW– Magnitude: proportional to the length of arrowMagnitude: proportional to the length of arrow– Direction: defined by angle between reference axis and line of Direction: defined by angle between reference axis and line of

action of the arrowaction of the arrow– Sense: indicated by the arrowheadSense: indicated by the arrowhead

In text, vector symbolized by boldface type, In text, vector symbolized by boldface type, AA, and its , and its magnitude (always positive) by |magnitude (always positive) by |A A | or simply | or simply AA

In slides, vector symbolized by boldface type, In slides, vector symbolized by boldface type, AA, and , and magnitude by regular type, Amagnitude by regular type, A

In handwritten work, vector represented by letter with an In handwritten work, vector represented by letter with an arrow over it, its magnitude by letter enclosed in absolute arrow over it, its magnitude by letter enclosed in absolute value symbol or by letter itself value symbol or by letter itself SHOWSHOW

Obey parallelogram law of additionObey parallelogram law of addition Examples: position, force, momentExamples: position, force, moment

Multiplication and division of a Multiplication and division of a vector by a scalarvector by a scalar

Product of a vector, Product of a vector, AA, and a scalar, a, and a scalar, a– Vector, aVector, aAA– Magnitude, aAMagnitude, aA– Sense of aSense of aAA is the same as is the same as AA provided a is positive, it is provided a is positive, it is

opposite to opposite to AA if a is negative if a is negative Division can be converted to multiplication and then Division can be converted to multiplication and then

laws of multiplication applied, laws of multiplication applied, AA/a = (1/a) /a = (1/a) AA, a≠0, a≠0 Product is associative with respect to scalar Product is associative with respect to scalar

multiplicationmultiplicationa(ba(bAA) = (ab)) = (ab)AA

Product is distributive with respect to scalar additionProduct is distributive with respect to scalar addition(a + b)(a + b)AA = a = aAA + b + bAA

Product is distributive with respect to vector additionProduct is distributive with respect to vector additiona(a(AA + + BB) = a) = aAA + a + aBB

Vector additionVector addition

Parallelogram law Parallelogram law SHOWSHOW– Join the tailsJoin the tails– Draw parallel dashed lines from the head of each vector to the Draw parallel dashed lines from the head of each vector to the

intersection at a common pointintersection at a common point– Resultant, Resultant, RR, is the diagonal of the parallelogram (extends from , is the diagonal of the parallelogram (extends from

the tails of the two vectors to the intersection of the dashed lines)the tails of the two vectors to the intersection of the dashed lines)– Special case – two vectors collinear – parallelogram law reduces Special case – two vectors collinear – parallelogram law reduces

to an algebraic or scalar addition, R = A + Bto an algebraic or scalar addition, R = A + B Triangular construction (head-to-tail fashion) Triangular construction (head-to-tail fashion) SHOWSHOW

– Adding Adding BB to to AA– Connect the tail of vector Connect the tail of vector BB to the head of vector to the head of vector AA– Resultant, Resultant, RR, extends from the tail of vector , extends from the tail of vector AA to the head of to the head of

vector vector BB Vector addition is commutative, Vector addition is commutative, AA + + BB = = BB + + AA Vector addition is associative, (Vector addition is associative, (AA + + BB) + ) + DD = = AA + ( + (BB + + DD))

Vector subtractionVector subtraction

Difference between two vectors Difference between two vectors AA and and BB– Subtraction defined as a special case of additionSubtraction defined as a special case of addition– R’R’ = = AA – – BB = = AA + (- + (-BB) = ) = AA + (-1) + (-1)BB

SHOWSHOW

Resolution of a vectorResolution of a vector

Vector may be resolved into two “components” having Vector may be resolved into two “components” having known lines of action by using the parallelogram lawknown lines of action by using the parallelogram law– SHOWSHOW– Starting at the head of Starting at the head of RR, extend dashed line parallel to , extend dashed line parallel to

a until it intersects b, likewise dashed line parallel to b a until it intersects b, likewise dashed line parallel to b until it intersects auntil it intersects a

– Two components Two components AA and and BB are then drawn such that they are then drawn such that they extend from the tail of extend from the tail of RR to the points of intersection to the points of intersection

– RR is resolved into the vector components is resolved into the vector components AA and and BB

Problems in Statics involving force Problems in Statics involving force (a vector quantity)(a vector quantity)

Find the resultant force, knowing its componentsFind the resultant force, knowing its components Resolve a known force into its componentsResolve a known force into its components

Procedure for analysis – addition of Procedure for analysis – addition of forcesforces

Apply parallelogram lawApply parallelogram law– Sketch vector addition using parallelogram lawSketch vector addition using parallelogram law– Determine interior angles from geometry of the problem Determine interior angles from geometry of the problem

(recall sum total of interior angles of parallelogram = (recall sum total of interior angles of parallelogram = 360°)360°)

– Label known angles and known forcesLabel known angles and known forces– Redraw half portion of constructed parallelogram to Redraw half portion of constructed parallelogram to

show triangular head-to-tail addition componentsshow triangular head-to-tail addition components– Unknowns can be determined from known data on Unknowns can be determined from known data on

triangle and use of trigonometrytriangle and use of trigonometry– If no 90° angle, law of sines and/or law of cosines may be If no 90° angle, law of sines and/or law of cosines may be

used used SHOWSHOW– EXAMPLES (pgs 29-32)EXAMPLES (pgs 29-32)

Vector components parallel to x and Vector components parallel to x and y axes (Cartesian components)y axes (Cartesian components)

FF can be resolved into its vector components can be resolved into its vector components FFxx and and FFyy parallel to parallel to the x and y axes, the x and y axes, F = FF = Fxx + + FFyy

Unit vectors Unit vectors ii and and jj designate directions along the x and y axes designate directions along the x and y axes– ii and and jj vectors have a dimensionless magnitude of unity vectors have a dimensionless magnitude of unity– Their direction will be described analytically by a “+” or “-” depending on Their direction will be described analytically by a “+” or “-” depending on

whether they are pointing along the positive or negative x or y axiswhether they are pointing along the positive or negative x or y axis SHOWSHOW

– FF = F = Fxxi i + F+ Fyyj j (Cartesian vector form)(Cartesian vector form)– F’F’ = F’ = F’xx(-(-ii) + F’) + F’yy(-(-jj) = - F’) = - F’xx((ii) - F’) - F’yy((jj))– The magnitude of each component of The magnitude of each component of FF is always a positive quantity, is always a positive quantity,

represented by the scalars Frepresented by the scalars Fxx and F and Fyy

– The magnitude of The magnitude of FF is given in terms of its components by the is given in terms of its components by the Pythagorean theorem,Pythagorean theorem,

– The direction angle The direction angle θθ, which specifies the orientation of the force, is , which specifies the orientation of the force, is determined from trigonometry, determined from trigonometry,

From this point forwardFrom this point forward i i and and jj will simply be written in regular type will simply be written in regular type since by definition they are vectorssince by definition they are vectors

22yx FFF

x

y

F

F1tan

Addition of vectors in terms of their Addition of vectors in terms of their (Cartesian) components(Cartesian) components

FF11 = (F = (F1x1xi + Fi + F1y1yj), j), FF22 = (F = (F2x2xi + Fi + F2y2yj), j), FF33 = (F = (F3x3xi + Fi + F3y3yj)j)

FFRR = F = F11 ++ F F22 ++ F F33

FFRR = (F = (F1x1xi + Fi + F1y1yj) + (Fj) + (F2x2xi + Fi + F2y2yj) + (Fj) + (F3x3xi + Fi + F3y3yj)j)

FFRR = F = F1x1xi + Fi + F2x2xi + Fi + F3x3xi + Fi + F1y1yj + Fj + F2y2yj + Fj + F3y3yjj

FFRR = (F = (F1x1x + F + F2x2x + F + F3x3x) i + (F) i + (F1y1y + F + F2y2y + F + F3y3y) j) j

FFRR = F = FRxRxi + Fi + FRyRyjj

– FFRx Rx = F= F1x1x + F + F2x2x + F + F3x3x or F or FRx Rx = = ∑ ∑ FFxx

– FFRy Ry = F= F1y1y + F + F2y2y + F + F3y3y or F or FRy Ry = = ∑ ∑ FFyy

EXAMPLES (pgs 40-43)EXAMPLES (pgs 40-43)

Rectangular components of a 3-D Rectangular components of a 3-D vectorvector

Assume right-handed coordinate systemAssume right-handed coordinate system AA = = AAxx + + AAyy + + AAzz SHOWSHOW

Cartesian vector notation, Cartesian vector notation, AA = A = Axxi + Ai + Ayyj j + A+ Azzkk

Magnitude of Cartesian vectorMagnitude of Cartesian vector– A = (AA = (Axx

22 + A + Ayy22 + A + Azz

22))1/21/2

– SHOWSHOW Direction of Cartesian vectorDirection of Cartesian vector

– SHOWSHOW

– cos cos αα = A = Axx/A, cos /A, cos ββ = A = Ayy/A, cos /A, cos γγ = A = Azz/A/A

Unit vectorUnit vector

Unit vector has a magnitude of 1 and specifies a Unit vector has a magnitude of 1 and specifies a directiondirection

If a unit vector If a unit vector uuAA and a vector and a vector AA have the same have the same direction direction →→ AA can be written as the product of its can be written as the product of its magnitude A and the unit vector magnitude A and the unit vector uuAA, , AA = A = A uuAA

– uuAA (dimensionless) defines the direction and sense of (dimensionless) defines the direction and sense of AA

– A (has dimensions) defines the magnitude of A (has dimensions) defines the magnitude of AA A unit vector having the same direction as A unit vector having the same direction as AA is is

represented by represented by uuAA = = AA/A/A

Using the unit vector to obtain the Using the unit vector to obtain the direction cosines direction cosines

A unit vector in the direction of A unit vector in the direction of AA ( (AA = A = Axxi + Ai + Ayyj + Aj + Azzk)k)

uuAA = = AA/A = /A = AAxx/A i + A/A i + Ayy/A j + A/A j + Azz/A k/A k

uuAA = = AA/A = u/A = uAAxxi + i + uuAAyyj + j + uuAAzzkk

((AAxx/A = /A = uuAAxx = = cos cos αα, A, Ayy/A = /A = uuAAyy = = cos cos ββ, A, Azz/A = /A = uuAAzz = = cos cos γγ))

uuAA = = AA/A = cos /A = cos αα i + cos i + cos ββ j + cos j + cos γγ k k

((uuAA has a magnitude 1 has a magnitude 1 and recalling that and recalling that A = (AA = (Axx22 + A + Ayy

22 + A + Azz22))1/21/2))

1 = (cos1 = (cos 2 2 αα + + coscos2 2 ββ + + coscos2 2 γγ))1/21/2

Squaring both sidesSquaring both sides1 = cos1 = cos

2 2 αα + + coscos2 2 ββ + + coscos2 2 γγ

Equation can be used to determine one of the coordinate Equation can be used to determine one of the coordinate direction angles if the other two are knowndirection angles if the other two are known

Addition and subtraction of 3-D Addition and subtraction of 3-D Cartesian vectorsCartesian vectors

AA = (A = (Axxi + Ai + Ayyj + Aj + AZZk), k), BB = (B = (Bxxi + Bi + Byyj + Bj + BZZk)k)

RR = = AA + + BB = (A = (Axx + B + Bxx) i + (A) i + (Ayy + B + Byy) j + (A) j + (Azz + B + Bzz) k) k

R’R’ = = AA - - BB = (A = (Axx - B - Bxx) i + (A) i + (Ayy - B - Byy) j + (A) j + (Azz - B - Bzz) k) k

FFRR = = ∑ ∑ FF = = ∑∑ F Fxxi + i + ∑∑ F Fyyj + j + ∑∑ F Fzzkk EXAMPLES (pgs 52-55)EXAMPLES (pgs 52-55)

Position vectorPosition vector

If If rr extends from point A (x extends from point A (xAA, y, yAA, z, zAA) to point B (x) to point B (xBB, y, yBB, , zzBB))– SHOWSHOW

– rrABAB = (x = (xBB – x – xAA) i + (y) i + (yBB – y – yAA) j + (z) j + (zBB – z – zAA) k) k

The i, j, k components of the position vector The i, j, k components of the position vector rrABAB may may be formed by taking the coordinates at the head of be formed by taking the coordinates at the head of the vector (point B, (xthe vector (point B, (xBB, y, yBB, z, zBB)) and subtracting the )) and subtracting the corresponding coordinates of the tail of the vector corresponding coordinates of the tail of the vector (point A, (x(point A, (xAA, y, yAA, z, zAA))))

Force vector directed along a lineForce vector directed along a line

The direction of a force can be specified by two points The direction of a force can be specified by two points through which its line of action passesthrough which its line of action passes

Formulate Formulate FF in a Cartesian vector form, realizing it has in a Cartesian vector form, realizing it has the same direction as the position vector the same direction as the position vector rr directed directed from point A to point B (A and B are points on a cord from point A to point B (A and B are points on a cord along along FF))

FF = F = Fuu = F ( = F (rr/r)/r) Procedure for analysisProcedure for analysis

– Determine position vector Determine position vector rr directed from A to B, and directed from A to B, and compute its magnitude rcompute its magnitude r

– Determine the unit vector Determine the unit vector uu = = rr/r which defines the /r which defines the direction of both direction of both rr and and FF

– Determine Determine FF by combining its magnitude F and direction by combining its magnitude F and direction uu

FF = F = Fuu EXAMPLES (pgs 65-68)EXAMPLES (pgs 65-68)

Dot productDot product

Dot product of vectors Dot product of vectors AA and and BB, , AA∙∙BB, “, “AA dot dot BB””– AA∙∙BB = AB cos = AB cos θθ (where 0°≤ (where 0°≤θ≤θ≤180°), 180°), SHOWSHOW– The result is a scalar, not a vectorThe result is a scalar, not a vector– The dot product is also referred to as the scalar product of The dot product is also referred to as the scalar product of

vectorsvectors Applicable laws of operationApplicable laws of operation

– Commutative law: Commutative law: AA∙∙BB = = BB∙∙AA– Multiplication by a scalar: a (Multiplication by a scalar: a (AA∙∙BB) = (a) = (aAA)∙)∙BB = = AA∙(a∙(aBB) = ) =

((AA∙∙BB) a) a– Distributive law: Distributive law: AA∙(∙(BB + + DD) = () = (AA∙∙BB) + () + (AA∙∙DD))

Cartesian vector formulationCartesian vector formulation– i∙i = (1)(1) cos 0° = 1i∙i = (1)(1) cos 0° = 1– i∙j = (1)(1) cos 90° = 0i∙j = (1)(1) cos 90° = 0– i∙k = (1)(1) cos 90° = 0i∙k = (1)(1) cos 90° = 0– Similarly, j∙j = 1, k∙k = 1, j∙k = 0Similarly, j∙j = 1, k∙k = 1, j∙k = 0

Dot product of two general vectorsDot product of two general vectors

AA∙∙BB = (A = (Axxi + Ai + Ayyj + Aj + AZZk)∙(Bk)∙(Bxxi + Bi + Byyj + Bj + BZZk)k)

= A= AxxBBxx (i∙i) + A (i∙i) + AxxBByy (i∙j) + A (i∙j) + AxxBBZZ (i∙k) (i∙k)

+ A+ AyyBBxx (j∙i) + A (j∙i) + AyyBByy (j∙j) + A (j∙j) + AyyBBzz (j∙k) (j∙k)

+ A+ AzzBBxx (k∙i) + A (k∙i) + AzzBByy (k∙j) + A (k∙j) + AzzBBzz (k∙k) (k∙k)

= A= AxxBBxx + A + AyyBByy + A + AzzBBzz

Applications of dot productApplications of dot product

Determining the angle formed between two vectors or Determining the angle formed between two vectors or intersecting linesintersecting lines(recall (recall AA∙∙BB = AB cos = AB cos θθ))

θθ = cos = cos-1-1 ( (AA∙∙BB/AB), 0°≤/AB), 0°≤θ≤θ≤180°180°

AA∙∙BB is computed from is computed from AA∙∙BB = A = AxxBBxx + A + AyyBByy + A + AzzBBzz

if if AA∙∙BB = 0 = 0 → → θθ = 90° = 90° → → AA is perpendicular to is perpendicular to BB

Applications of dot product Applications of dot product (continued)(continued)

Components of a vector parallel and perpendicular to a line (aa’)Components of a vector parallel and perpendicular to a line (aa’)– SHOWSHOW– ParallelParallel

For For AAp p (projection of (projection of AA onto line aa’) onto line aa’) →→ A App = A cos = A cos θθ Direction of line specified by the unit vector Direction of line specified by the unit vector uu (u=1) (u=1)

AApp = A cos = A cos θθ = (A)(1) cos = (A)(1) cos θθ = = AA∙∙u [u [and and AApp = ( = (AA∙∙uu)()(uu)])]

The scalar projection of The scalar projection of AA along a line is determined from the dot product along a line is determined from the dot product of of AA and the unit vector and the unit vector uu which defines the direction of the line which defines the direction of the line

If AIf App is positive, then is positive, then AApp has a sense which is the same as has a sense which is the same as uu

If AIf App is negative, then is negative, then AApp has the opposite sense to has the opposite sense to uu

– PerpendicularPerpendicular AA = = AApp + + AAnn → → AAnn = = AA - - AApp

AAnn → → AAnn = A sin = A sin θθ, , θθ = cos = cos-1-1 ( (AA∙∙uu/A)/A)

Or by Pythagorean Theorem AOr by Pythagorean Theorem Ann = (A = (A22 – A – App22))1/21/2

EXAMPLES (pg 76-80)EXAMPLES (pg 76-80)