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Vector AlgebraPrepared byMARINETTE RIVERA APAOApril 19, 2012
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Vector Algebra: Graphical MethodsTriangle (Tail-to-Head)
MethodScale: 1 div. = 3 m
Given:d1 = 9 m, Ed2 = 12 m, NdR = ? m
dR = 5 div dR = 15 m, 530
Use a protractor to measure the angle (from the x-axis)
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Vector Algebra: Graphical MethodsRight Angle (Tail-to-Tail)
MethodScale: 1 div. = 3 m
Given:d1 = 9 m, 00d2 = 12 m,900dR = ? m
dR = 5 div dR = 15 m, 530
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Vector Algebra: Graphical MethodsParallelogram (Tail-to-Tail)
MethodScale: 1 div. = 5 N
Given:F1 = 15 N, EF2 = 20 N, 450FR = ? N
FR = diagonal line of the parallelogramFR = 6.5 div FR = 32.5N,
250
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Vector Algebra: Graphical MethodsPolygon (Tail-to-Head)
MethodThe polygon method is used when adding > 2
vectors.Given:d1 = 4 m, Ed2 = 8 m, Nd3 = 6 m, 2100d4 = 3 m, SdR = ?
mScale: 1div = 1 mdR = 2.3 div dR = 2.3 m, 1200 ordR = 2.3 m, 600 N
of W
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Vector Algebra: Analytical MethodsPythagorean TheoremUsed when
adding 2 vectors that are 900 from each other
Equation: orc = a2 + b2 Exercise 1d1 = 9 m, Ed2 = 12 m, N
dR b = d2 a = d1 (
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Vector Algebra: Analytical MethodsCosine Law (magnitude
only)Used when adding 2 vectors that are < or > 900 from each
other
Equation:
a, b and c are sides. C is the angle opposite side c
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Vector Algebra: Analytical MethodsSine Law (direction/angle)Used
when adding 2 vectors that are < or > 900 from each other
(Equation: FR = 32.39N, 260
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Vector Algebra: Analytical MethodsComponent MethodUsed when
adding more than 2 vectorsStepsDetermine the x and y components of
each vector.Find the sum of the components along each
axis.Determine the quadrant where the Resultant Vector is expected
by illustrating the sums in the Cartesian planeCompute the
RESULTANT VECTOR using the Pythagorean Theorem for the magnitude
and tangent function for the direction
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Resolution of a VectorDetermining the components of a vectorA
vector has many possible pairs of components, but the easiest to
determine are the components that lie along the x and the y
axes.
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Example: Find the x and y components of a 20 N force directed at
500 Resolution of a VectorDetermining the x and y components of a
vectorFX = Fcos = (20N) cos500 = (20N)(0.6428) = 12.86 NFY = Fsin =
(20N) sin500 = (20N)(0.766) = 15.32 N
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Example: Find the resultant of the following vectors; v1=2.0 m/s
E, v2= 4.0 m/s N, v3 = 5.0 m/s 300 N of W v4 = 3.0 m/s 2400
Vector Algebra: Component MethodS1: Determine the x and y
components of each vectorS2: Find the sum along the x and y
VectorX - ComponentY - Componentv1= 2 m/s , Ev1x= v1cos00 = (2
m/s )(1) = 2.00 m/sv1y= v1sin00 = (2 m/s )(0) = 0V2= 4 m/s , Nv2x=
v2cos900 = (4 m/s )(0) = 0v2y= v2sin900 = (4 m/s )(1) = 4.00 m/sv3=
5 m/s , 300 N of Wv3x= v3cos300 = (5 m/s )(-0.866) = -4.33 m/sv3y=
v3sin300 = (5 m/s )(0.5) = 2.50 m/sv4= 3 m/s , 2400v4x= v4cos2400 =
(3 m/s )(-0.5) = -1.50 m/sv4y= v1sin2400 = (3 m/s )(-0.866) = -2.59
m/sSum ()Vx =2+0-4.33-1.5 = -3.83 m/sVy =0+4+2.5-2.59 = 3.91
m/s
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S3: Diagram X and Y in the Cartesian Plane Vector Algebra:
Component MethodS4: Determine the Resultant VectorVR = x2 + y2 =
(-3.83m/s)2 + (3.91 m/s)2 = 14.67 m2/s2 + 15.29 m2/s2 = 29.96 m2/s2
= 5.47 m/stan = y/x = 3.91m/s -3.83m/s = -1.02 = tan-1-1.02 = -460
VR = 5.5 m/s, 460 N of W
