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AP Calculus BC – Chapter 10 Parametric, Vector, and Polar Functions 10.2: Vectors in the Plane

AP Calculus BC – Chapter 10 Parametric, Vector, and Polar Functions 10.2: Vectors in the Plane. Goals : Represent vectors in the form <a, b> and perform algebraic computations involving vectors. Two-Dimensional Vector:.

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AP Calculus BC – Chapter 10 Parametric, Vector, and Polar Functions 10.2: Vectors in the Plane

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  1. AP Calculus BC – Chapter 10Parametric, Vector, and Polar Functions 10.2: Vectors in the Plane Goals: Represent vectors in the form <a, b> and perform algebraic computations involving vectors.

  2. Two-Dimensional Vector: A two-dimensional vector v is an ordered pair of real numbers, denoted in component form as a, b. The numbers a and b are the components of the vector v. The standard representation of the vector a, b isthe arrow from the origin to the point (a, b). The magnitude (or absolute value) of v, denoted |v|, is the length of the arrow, and the direction of v is the direction in which the arrow is pointing. The vector 0 = 0, 0, called the zero vector, has zero length and no direction.

  3. Magnitude and Direction: Magnitude of a Vector: The magnitude or absolute value of the vector a, b is the nonnegative real number Direction Angle of a Vector: The direction angle of a nonzero vector v is the smallest nonnegative angle formed with the positive x-axis as the initial ray and the standard representation of v as the terminal ray.

  4. Vector Operations: The algebra of vectors sometimes involves working with vectors and numbers at the same time. In this context, we refer to the numbers as scalars. The two most basic algebraic operations involving vectors are vector addition (adding a vector to a vector) and scalar multiplication (multiplying a vector by a number).

  5. Vector Operations: Let u= u1, u2 and v= v1, v2 be vectors and let k be a real number (scalar). The sum (or resultant) of the vectors u and v is the vector u + v= u1+v1 ,u2+ v2 The product of the scalar k and the vector u is ku=ku1 ,u2= ku1 ,ku2 The opposite of a vectorv is –v =(-1)v. Vector subtraction is defined by u - v=u+(-v).

  6. Properties of Vector Operations: Let u, v, and w be vectors and let a and b be scalars. 1. u + v = v + u commutativity 2. (u + v) + w = u + (v + w) associativity 3. u + 0 = u additive identity 4. u + (-u) = 0 additive inverses 5. 0u = 0 scalar multiplication by 0 6. 1u = u scalar multiplicative identity

  7. Properties of Vector Operations: Let u, v, and w be vectors and let a and b be scalars. 7. (ab)u = a(bu) 8. a(u + v) = au + av 9. (a + b)u = au + bu

  8. Angle Between Two Vectors: The angle between two nonzero vectors u= u1, u2 and v= v1, v2 is given by

  9. Dot Product (Inner Product): Dot Product (Inner Product): The dot product (or inner product) u•v of vectors u= u1, u2 and v= v1, v2 is the number Corollary - Angle Between Two Vectors:The angle between nonzero vectors u and v is

  10. Assignment: • HW 10.2: #3-30 (every 3rd), 31, 32, 35, 39, 43, 47.

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