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11.1 Vectors in the plane

11.1 Vectors in the plane. Miss Battaglia - AP Calculus Objective: Determine position, velocity, and acceleration using vectors . Component form of a vector.

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11.1 Vectors in the plane

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  1. 11.1 Vectors in the plane Miss Battaglia- AP Calculus Objective: Determine position, velocity, and acceleration using vectors

  2. Component form of a vector • Quantities (such as force, velocity, and acceleration) involve magnitude and direction. A directed line segment is used to represent these quantities. The directed line segment has an initial point P and terminal point Q, and its length (or magnitude) is denoted .

  3. Component form of a vector • Directed line segments that have the same length and direction are equivalent. The set of all directed line segments that are equivalent to a given directed line segment is a vector in the plane.

  4. Vector representation by directed line segments • Let v be represented by the directed line segment from (0,0) to (3,2) and let u be represented by the directed line segment from (1,2) to (4,4). Show that v and u are equivalent.

  5. An introduction to vectors • If a vector starts at ( x1, y1) and terminates at ( x2, y2 ), then its components are • < x2 – x1, y2 – y1 > • The magnitude is the length of the vector.

  6. Find the component form for each vector. Find the magnitude of the vector. • Initial point of (2, 3) and terminal point of (7, 6) • Initial point of (3, 1) and terminal point of (2, - 3)

  7. Definitions of vector addition and scalar multiplication • Let and be vectors and let c be a scalar. • The vector sum of u and v is the vector • The scalar multiple of c and u is the vector • The negative of v is the vector • The difference of u and v is

  8. Vector operations • Given and , find each of the vectors. • ½v • w – v • v + 2w

  9. Properties of vector operations • Let u, v, and w be vectors in the plane and let c and d be scalars. • u + v = v + w • (u + v) + w = u + (v + w) • u + 0 = u • u + (-u) = 0 • c(du) = (cd)u • (c + d)u = cu + du • c(u + v) = cu + cv • l(u) = u, 0(u) = 0

  10. A car travels with a velocity vector given by: • where t is measured in seconds, and the vector components are measured in feet.If the initial position of the car is: • find the position of the car after 1 second.

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