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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
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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 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 Scalars vs. Vectors
Vectors • Have both a magnitude and direction • Represented by: • Arrow on a graph • Boldface print with an arrow a
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…
Section B: Vector Components • Corresponding Book Sections: • 3.2 • PA Assessment Anchors: • S11.C.3
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 components of the vector r
In other words… r ry rx
How do you find those scalar components? • Trigonometric relationships • Sine • Cosine • Tangent • SOH – CAH – TOA
The basics… Ax = A cos θ Ay = A sin θ
How do you determine the signs (+ or -) of vector components?
How do you determine the signs (+ or -) of vector components?
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!
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°
Section C: Drawing Vectors • Corresponding Book Sections: • 3.3 • PA Assessment Anchors: • S11.C.3
You can move vectors! These are all the same vector – you just cannot change the length or direction.
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.
But wait…it gets even better… C = A + B = B + A
This means that… = C = A + B C = B + A
Subtracting Vectors Graphically • Suppose we’re looking for: D = A – B • This really is equal to: D = A + (-B)
So, what does a negative vector look like… • The negative vector is simply the magnitude of the original vector pointing in the opposite direction
Back to the treasure hunt Find both the magnitude and direction of the resultant vector C.
Section D: Combining Vectors (Component Method) • Corresponding Book Sections: • 3.3 • PA Assessment Anchors: • S11.C.3
Adding vectors using components… Ax = A cos θ Ay = A sin θ • Remember that: • To find C (where C = A + B): • Cx = Ax + Bx • Cy = Ay + By
Adding vectors using components (continued)… • And then…
Subtracting vectors using components… • To find D (where D = A - B): • Dx = Ax - Bx • Dy = Ay - By
Subtracting vectors using components (continued)… • And then…
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 Position vs. Displacement Vectors
Practice Problem #2 • Now draw the vectors and their components for those four angles. • Determine if each component is positive or negative
