Vector Mathematics AP
Transcript of Vector Mathematics AP
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Vector Mathematics
Adding, Subtracting, Multiplying
and Dividing
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Why?
One can add 23 kg and 42 kg and get 65
kg.
However, one cannot add together 23 m/ssouth and 42 m/s southeast and get 65
m/s south-southeast.
Vectors addition takes into account adding
both magnitude and direction
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Words
Vector : A measured quantity with both
magnitude (the how big part) and direction
Scalar : A measured quantity withmagnitude only
Resultant Vector : The final vector of a
vector math problem
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³Math´ Coordinate System
(Direction)
0º
90º
180º
270º
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Polar Coordinate System
(Direction and Magnitude)
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Polar ³Math´ (Cartesian)
2 2
1
cossin
tan
r x y
x r
y r
y
x
UU
U
!
!!
¨ ¸! © ¹ª º
x
yr
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Vector addition
Two Ways:
1. Graphically: Draw vectors to scale, Tip
to Tail, and the resultant is the straightline from start to finish
2. Mathematically: Employ vector math
analysis to solve for the resultant
vector
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Graphically 2-D Right
A = 5.0 m @ 0°
B = 5.0 m @ 90°
Solve A + B
R
Start
R=7.1 m @ 45°
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Important
You can add vectors in any order and yield
the same resultant.
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Let¶s add the last one
mathematically The math you used previously doesn¶t
work (and I won¶t let you use the Law of
Sines or Cosines) or does it???
What we will do is break each vector into
components
The components are the x and y values of
the polar coordinate (go back 6 slides)
Check out the next slides«
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Components of Vectors
A = Ax + Ay
Ax =A cos
Ay = A sin
As long as you
draw the xcomponent first
A
Ax
Ay
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The Table Method
We will organize these components in a
table.
See the board for this part and next slide
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Table Method Equation
Add all X components together Final Rx
Add all Y components together Final Ry
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Subtracting Vectors
Simply add or subtract 180° (keep
between 0° and 360°) to the direction of
the vector being subtracted
You just ADD the OPPOSITE vector (there
is no subtraction in vector math)
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Subtracting Vectors
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Unit Vectors
A unit vector is a vector that has a
magnitude of 1, with no units.
Its only purpose is to point We will use i, j, k f or our unit vectors
i means x ± dir ection, j is y, and k is z
We also put little ³hats´ (^) on i, j, k to show that they ar e unit vectors (I will
boldf ace them)
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Unit Vectors for vectors A & B
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Unit Vectors
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Adding using unit vectors
R = A + B
R = (Ax + Bx )i + (Ay + By ) j + (Az + Bz )k
which becomesR = R x i + R y j + R z k
The magnitude of R is f ound by a pplying the
Pythagor ean theor em
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Multiplying Vectors (products)
3 ways
1. Scalar x Vector = Vector w/
magnitude multiplied by the value of
scalar
A = 5 m @ 30°
3A = 15m @ 30°
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Multiplying Vectors (products)
2. (vector) (vector) = Scalar
This is called the Scalar Product or the
Dot Product
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Dot Product Continued (see p. 25)
A
B
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Multiplying Vectors (products)
3. (vector) x (vector) = vector
This is called the vector product or the
cross product
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Cross Product Continued
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Cross Product Direction and
reverse
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Cross Product
You can also solve the Cross Product with
a matrix and unit vectors«check out the
board for this.