Vector Mathematics
description
Transcript of Vector Mathematics
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VectorMathematics
Physics 1
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Physical Quantities
A scalar quantity is expressed in terms of magnitude (amount) only.
Common examples include time, mass, volume, and temperature.
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Physical QuantitiesA vector quantity is expressed in
terms of both magnitude and direction.
Common examples include velocity, weight (force), and acceleration.
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Representing VectorsVector quantities can be graphically
represented using arrows.– magnitude = length of the arrow– direction = arrowhead
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Vectors All vectors have a head and a tail.
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Vector Addition
Vector quantities are added graphically by placing them head-to-tail.
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Head-to-Tail Method
1. Draw the first COMPONENT vector with the proper length and orientation.
2. Draw the second COMPONENT vector with the proper length and orientation starting from the head of the first component vector.
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Head-to-Tail Method
3. The RESULTANT (sum) vector is drawn starting at the tail of the first component vector and terminating at the head of the second component vector.
4. Measure the length and orientation of the resultant vector.
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South
East
Resultant
To add vectors, move tail to head and then draw resultant from original start to final point.
Resultant is (sqrt(2)) 45◦ south
of East
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South
East
Resultant
Vector addition is ‘commutative’ (can add vectors in either order)
Resultant is (sqrt(2)) 45◦ south
of East
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South
East
Resultant
Vector addition is ‘commutative’ (can add vectors in either order)
Resultant is (sqrt(2)) 45◦ south
of East
South
East
Resultant
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Co-linear vectors make a longer (or shorter) vector
Resultant is 3 magnitude South
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Co-linear vectors make a longer (or shorter) vector
Resultant is 3 magnitude South
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Nor
thN
orth
EastEast
Can add multiple vectors.Just draw ‘head to tail’ for each vector
Resultant is magnitude
45◦ North of East 2
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Nor
thSo
uth
EastEast
Adding vectors is commutative.
Resultant is magnitude
45◦ North of East
Nor
th
East
Nor
th
East
22
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Nor
th
South
East
WestResultant=0
Equal but opposite vectors cancel each other out
Resultant is 0.
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Vector Addition – same directionA + B = R
B
A
A B
R = A + B
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Vector Addition
• Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north?
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Vector Addition – Opposite direction(Vector Subtraction) .
A + (-B) = RA
B
-B
A
-B A + (-B) = R
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Vectors• The sum of two or more vectors is called the
resultant.
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Practice
Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html
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Polar VectorsEvery vector has a magnitude and
direction
direction anglemagnitude
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Right Triangles
SOH CAH TOA
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Vector ResolutionEvery vector quantity can be
resolved into perpendicular components.
Rectilinear (component) form of vector:
yx
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A
Ax
Ay
Vector Resolution
Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).
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Vector ResolutionIf these two components were
added together, the resultant would be equal to vector A.
A
Ax
Ay
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Vector ResolutionWhen resolving a vector graphically, first
construct the horizontal component (Ax). Then construct the vertical component (Ay).
Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived.
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Vector Resolution
Horizontal Component (Ax)
AAxcos
cosAAx
A
Ax
Ay
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Vector Resolution
Vertical Component (Ay)
sin yAA
sinyA A
A
Ax
Ay
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Drawing Directions
EX: 30° S of W– Start at west axis and move south 30 °– Degree is the angle between south and west
N
S
EW
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Vector ResolutionUse the sign
conventions for the x-y coordinate system to determine the direction of each component.
(+,+)(-,+)
(-,-) (+,-)
N
E
S
W
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Component Method1. Resolve all vectors into
horizontal and vertical components.
2. Find the sum of all horizontal components. Express as SX.
3. Find the sum of all vertical components. Express as SY.
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Component Method4. Construct a vector diagram using the
component sums. The resultant of this sum is vector A + B.
5. Find the magnitude of the resultant vector A + B using the Pythagorean Theorem.
6. Find the direction of the resultant vector A + B using the tangent of an angle .
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Adding “Oblique” Vectors
Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector
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Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
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Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
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Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors
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Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
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Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
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Using Calculator For Vectors
Can use the “Angle” button on TI-84 calculator to do vector mathematics
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Using Calculator for Vectors