Chapter 1 Topics in Analytic Geometry

••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 78that is normal to C and points in the same direction as T ′ (t). We call N(t) the principalunit normal vector to C at t, or more simply, the unit normal vector. Observe thatthe unit normal vector is defined only at points where T ′ (t) ≠ 0.In 2-space there are two unit vectors that are orthogonal to T(t), and in 3-space thereare infinitely such vectors.yCzT(t)CxT(t)yThere are two unit vectorsorthogonal to T(t).xThere are infinitely many unitvectors orthogonal to T(t).Example 4.26 Find T(t) and N(t) for the circular helixwhere a > 0.Solution .........x = acost, y = asint, z = ctExample 4.27 Find the unit tangent and principal unit normal vectors to the curve definedby r(t) = 〈t 2 ,t〉.Solution .........Computing T and N for Curves Parametrized by Arc LengthIn the case where r(s) ia parametrized by arc length, the procedures for computing theunit tangent vector T(s) and the unit normal vector N(s) are simpler than in the generalcase. For example, we showed in Theorem 4.9 that if s is an arc length parameter, then‖r ′ (s) = 1. Thus, Formula (4.36) for the unit tangent vector simplifies toT(s) = r ′ (s) (4.38)and consequently Formula (4.37) for the unit normal vector simplifies toN(s) = r′′ (s)‖r ′′ (s)‖(4.39)Example 4.28 The circle of radius a with counterclockwise orientation and centered at theorigin can be represented by the vector-valued functionr = acosti+asintj (0 ≤ t ≤ 2π)Parametrize this circle by arc length and find T(s) and N(s).Solution .........