Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 70Derivatives of Dot and Cross ProductsThe following rules provide a method for differentiating dot products in 2-space and 3-spaceand cross product in 3-space.ddt [r 1(t)·r 2 (t)] = r 1 (t)· dr 2dt + dr 1dt ·r 2(t) (4.11)ddt [r 1(t)×r 2 (t)] = r 1 (t)× dr 2dt + dr 1dt ×r 2(t) (4.12)Theorem 4.4 If r(t) is a differentiable vector-valued function in 2-space or 3-space and‖r(t)‖ is constant for all t, thenr(t)·r ′ (t) = 0 (4.13)that is, r(t) and r ′ (t) are orthogonal vectors for all t.Definite Integrals of Vector-Valued FunctionsIf r(t) is a vector-valued function that is continuous on the interval a ≤ t ≤ b, then wedefine the definite integral of r(t) over this interval as a limit of Riemann sums, that is,In general, we have∫ ba∫ ba(∫ br(t)dt =∫ bar(t)dt =(∫ br(t)dt =aalimmax △t k →0n∑r(t ∗ k)△t k (4.14)k=1) (∫ b)x(t)dt i+ y(t)dt j 2-space (4.15)a) (∫ b) (∫ b)x(t)dt i+ y(t)dt j+ z(t)dt k 3-space (4.16)aaExample 4.12 Let r(t) = t 2 i+e t j−(2cosπt)k. EvaluateSolution .........∫ 10r(t)dt.Rules of IntegrationAs with differentiating, many of the rules for integrating real-values functions have analogsfor vector-values functions.Theorem 4.5 (Rulesof Integration). Letr(t), r 1 (t), and r 2 (t) be vector-valued functionsin 2-space or 3-space that are continuous on the interval a ≤ t ≤ b, and let k be a scalar.Then the following rules of integration hold: