unit 2: vectors

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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