unit 2: vectors
DESCRIPTION
Unit 2: Vectors. Section A: Vectors vs. Scalars. Corresponding Book Sections: 2.1, 2.2, 3.1 PA Assessment Anchors: S11.C.3. Which is more specific?. Option A: The library is 0.5 mile from here Option B: The library is 0.5 mile to the northwest from here. Scalars Number Has Units - PowerPoint PPT PresentationTRANSCRIPT
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Unit 2: Vectors
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Section A: Vectors vs. Scalars
Corresponding Book Sections: 2.1, 2.2, 3.1
PA Assessment Anchors: S11.C.3
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Which is more specific?
Option A: The library is 0.5 mile from here
Option B: The library is 0.5 mile to the northwest from here
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Scalars vs. Vectors Scalars
Number Has Units Positive, Negative,
Zero
Ex: The library is 0.5 mile from here
Vectors Magnitude
Distance covered Direction
Ex: The library is 0.5 mile northwest from here
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Why is this important?
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Vectors
Have both a magnitude and direction
Represented by: Arrow on a graph Boldface print with an arrow a
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Back to the example…
The library is 0.5 mile to the northwest.
How do we actually get to the library? Probably not possible to walk in a
straight line…
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Section B: Vector Components
Corresponding Book Sections: 3.2
PA Assessment Anchors: S11.C.3
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Now explain how to get to the library…
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Vector Components
If we have a “resultant” vector r
We break a vector down into its components: x-direction: rx
y-direction: ry
These are called “scalar componentsof the vector r
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In other words…
r
rx
ry
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How do you find those scalar components? Trigonometric relationships
Sine Cosine Tangent
SOH – CAH – TOA
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The basics…
Ax = A cos θ
Ay = A sin θ
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To find the magnitude and direction given the components:
2 2x yA A A
1tan y
x
A
A
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How do you determine the signs (+ or -) of vector components?
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How do you determine the signs (+ or -) of vector components?
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Summary of those four pictures… To determine the sign of a vector
component: Look at the direction in which they
point If the component points in positive
direction, it is positive If the component points in negative
direction, it is negative THIS DOES NOT MEAN THE VECTOR IS
POSITIVE OR NEGATIVE!
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Practice Problem #1
The vector A has a magnitude of 7.25m
Find its components for: θ = 5.00° θ = 125° θ = 245° θ = 335°
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Section C: Drawing Vectors
Corresponding Book Sections: 3.3
PA Assessment Anchors: S11.C.3
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A picture…
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You can move vectors!
These are all the same vector – you just cannot change the length or direction.
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Adding Vectors Graphically
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The Vector Addition Rule…
To add the vectors A and B: Place the tail of B to the head of A.
C = A + B, is the vector extending from the tail of A to the head of B.
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But wait…it gets even better…
C = A + B = B + A
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This means that…
C = A + B C = B + A=
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Subtracting Vectors Graphically
Suppose we’re looking for:D = A – B
This really is equal to:D = A + (-B)
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So, what does a negative vector look like… The negative
vector is simply the magnitude of the original vector pointing in the opposite direction
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Back to the treasure hunt
Find both the magnitudeand direction of theresultant vector C.
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Section D: Combining Vectors (Component Method)
Corresponding Book Sections: 3.3
PA Assessment Anchors: S11.C.3
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Adding vectors using components…
Remember that:
To find C (where C = A + B): Cx = Ax + Bx
Cy = Ay + By
Ax = A cos θ
Ay = A sin θ
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Adding vectors using components (continued)…
And then…2 2x yC C C
1tan y
x
C
C
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Subtracting vectors using components…
To find D (where D = A - B): Dx = Ax - Bx
Dy = Ay - By
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Subtracting vectors using components (continued)…
And then…2 2x yD D D
1tan y
x
D
D
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Position vs. Displacement Vectors Position Vector
Indicated from the origin to the position in question
Ex: Where you are from the origin
Displacement Vector The change from
the initial position to the final position
Ex: Δr = rf – ri
This means that…rf = Δr + ri
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A displacement vector…
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Practice Problem #2
Now draw the vectors and their components for those four angles.
Determine if each component is positive or negative