Example Given: A = A x i + A y j + A Z k and B = B x i + B y j + B Z k Vector Addition Resultant R =...
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Transcript of Example Given: A = A x i + A y j + A Z k and B = B x i + B y j + B Z k Vector Addition Resultant R =...
ExampleGiven: A = Axi + Ayj + AZk
and B = Bxi + Byj + BZk
Vector AdditionResultant R = A + B
= (Ax + Bx)i + (Ay + By )j + (AZ + BZ) kVector SubstractionResultant R = A - B
= (Ax - Bx)i + (Ay - By )j + (AZ - BZ) k
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsConcurrent Force Systems
- Force resultant is the vector sum of all the forces in the system
FR = ∑F = ∑Fxi + ∑Fyj + ∑Fzk
where ∑Fx , ∑Fy and ∑Fz represent the algebraic sums of the x, y and z or i, j or k components of each force in the system
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
Force, F that the tie down rope exerts on the ground support at O is directed along the rope
Angles α, β and γ can be solved with axes x, y and z
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
Cosines of their values forms a unit vector u that acts in the direction of the rope
Force F has a magnitude of FF = Fu = Fcosαi + Fcosβj + Fcosγk
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsExample 2.8Express the force F as Cartesian
vector
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionSince two angles are specified, the third angle is found by
Two possibilities exit, namelyor
605.0cos
5.0707.05.01cos
145cos60coscos
1coscoscos
1
22
222
222
1205.0cos 1
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionBy inspection, α = 60° since Fx is in the +x
directionGiven F = 200N
F = Fcosαi + Fcosβj + Fcosγk = (200cos60°N)i + (200cos60°N)j
+ (200cos45°N)k = {100.0i + 100.0j + 141.4k}N
Checking:
N
FFFF zyx
2004.1410.1000.100 222
222
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors Example 2.9 Determine the magnitude and
coordinate direction angles of resultant force
acting on the ring
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionResultant force
FR = ∑F
= F1 + F2 = {60j + 80k}kN
+ {50i - 100j + 100k}kN = {50j -40k + 180k}kN
Magnitude of FR is found by
kN
FR
1910.191
1804050 222
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionUnit vector acting in the direction of FR
uFR = FR /FR
= (50/191.0)i + (40/191.0)j + (180/191.0)k
= 0.1617i - 0.2094j + 0.9422kSo that
cosα = 0.2617 α = 74.8° cos β = -0.2094 β = 102° cosγ = 0.9422 γ = 19.6°
*Note β > 90° since j component of uFR is negative
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsExample 2.10Express the force F1 as a Cartesian
vector.
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionThe angles of 60° and 45° are not
coordinate direction angles.
By two successive applications ofparallelogram law,
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionBy trigonometry,
F1z = 100sin60 °kN = 86.6kNF’ = 100cos60 °kN = 50kNF1x = 50cos45 °kN = 35.4kNF1y = 50sin45 °kN = 35.4kN
F1y has a direction defined by –j, Therefore
F1 = {35.4i – 35.4j + 86.6k}kN
SolutionChecking:
Unit vector acting in the direction of F1
u1 = F1 /F1
= (35.4/100)i - (35.4/100)j + (86.6/100)k
= 0.354i - 0.354j + 0.866k
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
N
FFFF zyx
1006.864.354.35 222
21
21
211
Solutionα1 = cos-1(0.354) = 69.3°
β1 = cos-1(-0.354) = 111°
γ1 = cos-1(0.866) = 30.0°
Using the same method, F2 = {106i + 184j - 212k}kN
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsExample 2.11Two forces act on the hook. Specify the coordinate direction angles of F2, so that the
resultant force FR acts along the positive y axis and has a magnitude of 800N.
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionCartesian vector formFR = F1 + F2
F1 = F1cosα1i + F1cosβ1j + F1cosγ1k
= (300cos45°N)i + (300cos60°N)j + (300cos120°N)k
= {212.1i + 150j - 150k}NF2 = F2xi + F2yj + F2zk
View Free Body Diagram
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionSince FR has a magnitude of 800N and actsin the +j directionFR = F1 + F2
800j = 212.1i + 150j - 150k + F2xi + F2yj + F2zk
800j = (212.1 + F2x)i + (150 + F2y)j + (- 50 + F2z)k
To satisfy the equation, the corresponding components on left and right sides must be equal
2.6 Addition and Subtraction of Cartesian
Vectors
2.6 Addition and Subtraction of Cartesian
VectorsSolutionHence,
0 = 212.1 + F2x F2x = -212.1N 800 = 150 + F2y F2y = 650N 0 = -150 + F2z F2z = 150N
Since magnitude of F2 and its components are known, α1 = cos-1(-212.1/700) = 108° β1 = cos-1(650/700) = 21.8° γ1 = cos-1(150/700) = 77.6°
x,y,z Coordinates- Right-handed coordinate system- Positive z axis points upwards, measuring the height of an object or the altitude of a point- Points are measured relative to the origin, O.
2.7 Position Vectors2.7 Position Vectors
x,y,z CoordinatesEg: For Point A, xA = +4m along the x axis, yA = -6m along the y axis and zA = -6m
along the z axis. Thus, A (4, 2, -6) Similarly, B (0, 2, 0) and C (6, -1, 4)
2.7 Position Vectors2.7 Position Vectors
Position Vector- Position vector r is defined as a fixed vector which locates a point in space relative to another point. Eg: If r extends from the origin, O to point P (x, y, z) then, in Cartesian vector form
r = xi + yj + zk
2.7 Position Vectors2.7 Position Vectors
Position VectorNote the head to tail vector addition of the three components
Start at origin O, one travels x in the +i direction,
y in the +j direction and z in the +k direction, arriving at point P (x, y, z)
2.7 Position Vectors2.7 Position Vectors
2.7 Position Vectors2.7 Position Vectors Position Vector
- Position vector maybe directed from point A to point B - Designated by r or rAB
Vector addition givesrA + r = rB
Solving r = rB – rA = (xB – xA)i + (yB – yA)j + (zB –zA)k
or r = (xB – xA)i + (yB – yA)j + (zB –zA)k
Position Vector- The i, j, k components of the positive vector r may be formed by taking the coordinates of the tail, A (xA, yA, zA) and subtract them from the head B (xB, yB, zB)
Note the head to tail vector addition of the three components
2.7 Position Vectors2.7 Position Vectors
2.7 Position Vectors2.7 Position Vectors
Length and direction of cable AB can be found by measuring A and B using the x, y, z axes
Position vector r can be established
Magnitude r represent the length of cable
2.7 Position Vectors2.7 Position Vectors
Angles, α, β and γ represent the direction of the cable
Unit vector, u = r/r
2.7 Position Vectors2.7 Position Vectors
Example 2.12An elastic rubber band is attached to points A and
B. Determine its length and
its direction measured from
A towards B.
2.7 Position Vectors2.7 Position Vectors
SolutionPosition vector
r = [-2m – 1m]i + [2m – 0]j + [3m – (-3m)]k = {-3i + 2j + 6k}m
Magnitude = length of the rubber band
Unit vector in the director of ru = r /r = -3/7i + 2/7j + 6/7k
mr 7623 222
View Free Body Diagram
2.7 Position Vectors2.7 Position Vectors
Solutionα = cos-1(-3/7) = 115°
β = cos-1(2/7) = 73.4°
γ = cos-1(6/7) = 31.0°
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
In 3D problems, direction of F is specified by 2 points, through which its line of action lies
F can be formulated as a Cartesian vector
F = F u = F (r/r)
Note that F has units of forces (N) unlike r, with units of length (m)
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
Force F acting along the chain can be presented as a Cartesian vector by- Establish x, y, z axes- Form a position vector r along length of chain
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
Unit vector, u = r/r that defines the direction of both the chain and the force
We get F = Fu
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
Example 2.13The man pulls on the cord with a force of 350N. Represent this force
acting on the support A, as a Cartesian vector and determine its direction.
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionEnd points of the cord are A (0m, 0m,
7.5m) and B (3m, -2m, 1.5m)r = (3m – 0m)i + (-2m – 0m)j + (1.5m –
7.5m)k= {3i – 2j – 6k}m
Magnitude = length of cord AB
Unit vector, u = r /r = 3/7i - 2/7j - 6/7k
mmmmr 7623 222
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionForce F has a magnitude of 350N, direction specified by u
F = Fu = 350N(3/7i - 2/7j - 6/7k) = {150i - 100j - 300k} N
α = cos-1(3/7) = 64.6° β = cos-1(-2/7) = 107° γ = cos-1(-6/7) = 149°
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
Example 2.14The circular plate is partially supported by the cable AB. If the force of the cable on
the hook at A is F = 500N, express F as a Cartesian vector.
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionEnd points of the cable are (0m, 0m, 2m)
and B (1.707m, 0.707m, 0m)
r = (1.707m – 0m)i + (0.707m – 0m)j + (0m – 2m)k
= {1.707i + 0.707j - 2k}mMagnitude = length of cable AB
mmmmr 723.22707.0707.1 222
SolutionUnit vector,
u = r /r = (1.707/2.723)i + (0.707/2.723)j – (2/2.723)k
= 0.6269i + 0.2597j – 0.7345kFor force F,
F = Fu = 500N(0.6269i + 0.2597j – 0.7345k) = {313i - 130j - 367k} N
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionChecking
Show that γ = 137° and indicate this angle on the diagram
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
N
F
500
367130313 222
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
Example 2.15The roof is supported by cables. If the cables exert FAB = 100N and FAC = 120N
on the wall hook at A, determine the magnitude of the resultant force acting at A.
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionrAB = (4m – 0m)i + (0m – 0m)j + (0m –
4m)k = {4i – 4k}m
FAB = 100N (rAB/r AB)
= 100N {(4/5.66)i - (4/5.66)k} = {70.7i - 70.7k} N
mmmrAB 66.544 22
View Free Body Diagram
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionrAC = (4m – 0m)i + (2m – 0m)j + (0m – 4m)k
= {4i + 2j – 4k}m
FAC = 120N (rAB/r AB)
= 120N {(4/6)i + (2/6)j - (4/6)k} = {80i + 40j – 80k} N
mmmmrAC 6424 222
2.8 Force Vector Directed along a Line
2.8 Force Vector Directed along a Line
SolutionFR = FAB + FAC
= {70.7i - 70.7k} N + {80i + 40j – 80k} N = {150.7i + 40j – 150.7k} N
Magnitude of FR
N
FR
217
7.150407.150 222
2.9 Dot Product2.9 Dot ProductDot product of vectors A and B is
written as A·B (Read A dot B)Define the magnitudes of A and B and
the angle between their tails A·B = AB cosθ where 0°≤ θ
≤180°Referred to as scalar
product of vectors as result is a scalar
2.9 Dot Product2.9 Dot Product
Laws of Operation1. Commutative law
A·B = B·A2. Multiplication by a scalar
a(A·B) = (aA)·B = A·(aB) = (A·B)a
3. Distribution lawA·(B + D) = (A·B) + (A·D)
2.9 Dot Product2.9 Dot Product
Cartesian Vector Formulation- Dot product of Cartesian unit vectorsEg: i·i = (1)(1)cos0° = 1 andi·j = (1)(1)cos90° = 0- Similarly
i·i = 1 j·j = 1 k·k = 1 i·j = 0 i·k = 1 j·k = 1
2.9 Dot Product2.9 Dot Product
Cartesian Vector Formulation- Dot product of 2 vectors A and BA·B = (Axi + Ayj + Azk)· (Bxi + Byj + Bzk)
= AxBx(i·i) + AxBy(i·j) + AxBz(i·k)
+ AyBx(j·i) + AyBy(j·j) + AyBz(j·k)
+ AzBx(k·i) + AzBy(k·j) + AzBz(k·k)
= AxBx + AyBy + AzBz
Note: since result is a scalar, be careful of including any unit vectors in the result
2.9 Dot Product2.9 Dot Product
Applications- The angle formed between two vectors or intersecting lines
θ = cos-1 [(A·B)/(AB)] 0°≤ θ ≤180°
Note: if A·B = 0, cos-10= 90°, A is perpendicular to B
2.9 Dot Product2.9 Dot ProductApplications
- The components of a vector parallel and perpendicular to a line- Component of A parallel or collinear with line aa’ is defined by A║ (projection of A onto the line)
A║ = A cos θ
- If direction of line is specified by unit vector u (u = 1),
A║ = A cos θ = A·u
2.9 Dot Product2.9 Dot Product
Applications- If A║ is positive, A║ has a directional sense same as u- If A║ is negative, A║ has a directional sense opposite to u- A║ expressed as a vector
A║ = A cos θ u = (A·u)u
ApplicationsFor component of A perpendicular to line aa’ 1. Since A = A║ + A┴,
then A┴ = A - A║
2. θ = cos-1 [(A·u)/(A)]then A┴ = Asinθ
3. If A║ is known, by Pythagorean Theorem
2.9 Dot Product2.9 Dot Product
2||
2 AAA
2.9 Dot Product2.9 Dot Product For angle θ between the
rope and the beam A, - Unit vectors along the beams, uA = rA/rA
- Unit vectors along the ropes, ur=rr/rr
- Angle θ = cos-1 (rA.rr/rArr)
= cos-1 (uA· ur)
2.9 Dot Product2.9 Dot Product
For projection of the force along the beam A - Define direction of the beam
uA = rA/rA
- Force as a Cartesian vector
F = F(rr/rr) = Fur
- Dot product F║ = F║·uA
2.9 Dot Product2.9 Dot Product
Example 2.16The frame is subjected to a horizontal force F = {300j} N. Determine the components of this force parallel and perpendicular to the member AB.
2.9 Dot Product2.9 Dot Product
SolutionSince
Then
N
kjijuF
FF
kji
kjirr
u
B
AB
B
BB
1.257
)429.0)(0()857.0)(300()286.0)(0(
429.0857.0286.0300.
cos
429.0857.0286.0
362
362222
2.9 Dot Product2.9 Dot ProductSolutionSince result is a positive scalar,FAB has the same sense of
direction as uB. Express in Cartesian form
Perpendicular component
NkjikjijFFF
Nkji
kjiN
uFF
AB
ABABAB
}110805.73{)1102205.73(300
}1102205.73{
429.0857.0286.01.257
2.9 Dot Product2.9 Dot ProductSolutionMagnitude can be determined From F┴ or from Pythagorean
Theorem
N
NN
FFF AB
155
1.257300 22
22
2.9 Dot Product2.9 Dot Product
Example 2.17The pipe is subjected to F = 800N. Determine the angle θ between F and pipe segment BA, and the magnitudes of the components of F, which are parallel and perpendicular to BA.
2.9 Dot Product2.9 Dot Product
SolutionFor angle θrBA = {-2i - 2j + 1k}m
rBC = {- 3j + 1k}m
Thus,
5.42
7379.0
103
113202cos
BCBA
BCBA
rr
rr
View Free Body Diagram
2.9 Dot Product2.9 Dot Product
SolutionComponents of F
N
kjikj
uFF
kji
kjirr
u
BAB
AB
ABAB
590
3.840.5060
31
32
32
0.2539.758
.
31
32
32
3)122(
2.9 Dot Product2.9 Dot Product
SolutionChecking from trigonometry,
Magnitude can be determined From F┴
N
N
FFAB
540
5.42cos800
cos
NFF 5405.42sin800sin
2.9 Dot Product2.9 Dot Product
SolutionMagnitude can be determined from F┴ or
from Pythagorean Theorem
N
FFF AB
540
590800 22
22
Chapter SummaryChapter Summary
Parallelogram LawAddition of two vectorsComponents form the side and
resultant form the diagonal of the parallelogram
To obtain resultant, use tip to tail addition by triangle rule
To obtain magnitudes and directions, use Law of Cosines and Law of Sines
Chapter SummaryChapter Summary
Cartesian Vectors Vector F resolved into Cartesian vector
formF = Fxi + Fyj + Fzk
Magnitude of F
Coordinate direction angles α, β and γ are determined by the formulation of the unit vector in the direction of F
u = (Fx/F)i + (Fy/F)j + (Fz/F)k
222zyx FFFF
Chapter SummaryChapter Summary
Cartesian Vectors Components of u represent cosα, cosβ and
cosγ These angles are related by
cos2α + cos2β + cos2γ = 1
Force and Position Vectors Position Vector is directed between 2 points Formulated by distance and direction moved
along the x, y and z axes from tail to tip
Chapter SummaryChapter Summary
Force and Position Vectors For line of action through the two
points, it acts in the same direction of u as the position vector
Force expressed as a Cartesian vectorF = Fu = F(r/r)
Dot Product Dot product between two vectors A and
B A·B = AB cosθ
Chapter SummaryChapter Summary
Dot Product Dot product between two vectors A and B
(vectors expressed as Cartesian form)A·B = AxBx + AyBy + AzBz
For angle between the tails of two vectors θ = cos-1 [(A·B)/(AB)]
For projected component of A onto an axis defined by its unit vector u
A = A cos θ = A·u
Chapter ReviewChapter Review
Chapter ReviewChapter Review
Chapter ReviewChapter Review
Chapter ReviewChapter Review
Chapter ReviewChapter Review
Chapter ReviewChapter Review