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AP Physics C Mrs. Coyle

Vectors and Vector Operations. AP Physics C Mrs. Coyle. Coordinate Systems Vectors and Scalars Properties of Vectors Unit Vectors. Coordinate Systems. Cartesian (rectangular) Coordinates (x,y) Polar Coordinates (r, θ )

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AP Physics C Mrs. Coyle

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  1. Vectors and Vector Operations AP Physics C Mrs. Coyle

  2. Coordinate Systems • Vectors and Scalars • Properties of Vectors • Unit Vectors

  3. Coordinate Systems • Cartesian (rectangular) Coordinates (x,y) • Polar Coordinates (r, θ) • θ is taken to be positive counterclockwise from the +x axis.

  4. Vectors and Scalars • Scalars- magnitude only • Vectors- magnitude and direction

  5. Equality of Vectors • Two vectors are equal if they have the same magnitude and direction.

  6. Addition of Vectors • Graphical • Algebraic • Resultant: sum of vectors

  7. Properties of Vector Addition • Commutative Property of Addition • A + B = B + A • Associative Property of Addition • (A + B) + C = A + (B + C)

  8. Graphical Addition of Vectors • Head-to-Tail Method • Parallelogram Method

  9. Graphical Addition of Vectors Head-to-Tail Method • Vectors are moved parallel to themselves so that they are positioned in such a way that the head of one is adjacent to the tail of the other. • The resultant is drawn by starting at the first tail (loose tail) and ending (arrow head pointed) at the last head (loose head). B A Resultant

  10. Graphical Addition of Vectors • Parallelogram Method • The vectors are placed tail to tail forming a rectangle. • The diagonal that starts at the joint tails has its tail at the joint tails) is the resultant. A Resultant B

  11. Graphical Vector Subtraction When subtracting A-B : • Invert vector B to get -B • Add A+(-B) normally

  12. Algebraic Addition of Vectors-Component Method 1)Find x and y components of each vector. ax = acosθ ay = a sinθ

  13. Component Method Cont’d 2)Add x and y components. 3)Use the Pythagorean Theorem to find the magnitude of the resultant. 4)Use q=tan-1 |Y | to find the direction X with respect to the x-axis.

  14. Unit Vectors: î, ĵ, k • Dimensionless vector with a magnitude of 1. • They specify direction x, y, z • Example: A= 2 î + 3 ĵ - 6k

  15. Example 1 • Add the vectors: A= 10 î - 1 ĵ -6k B= - 6î + 5 ĵ +6k Give the components of the resultant vector, its magnitude and its direction with respect to the x-axis. Answer: R= 4î + 4ĵ, 5.7, 45 deg above +x axis

  16. Example 2 The position vector as a function of time for an object is given by r(t)= 2 î + 3t ĵ - 6k, r is in meters and t is in seconds. Evaluate dr/dt and explain what is its significance?

  17. Example 3 • These are instructions for finding a treasure : Go 75.0 paces at 240°, turn to 135° and walk 125 paces, then travel 100 paces at 160°. The angles are measured ccw from the east, the +x direction. Determine the resultant displacement from the starting point. • Answer: 227 paces at 165°

  18. Useful Link • http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html#veccon

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