Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 76Solution .........Example 4.23 Find the arc length parametrization of the linex = 2t+1, y = 3t−2that has the same orientation as the given line and uses (1,−2) as the reference point.Solution .........Properties of Arc Length ParametrizationsTheorem 4.9(a) If C is the graph of a smooth vector-valued function r(t) in 2-space or 3-space, wheret is a general parameter, and if s is the arc length parameter for C defined by Formula(4.27), then for every value of t the tangent vector has lengthdr∥dt∥ = ds(4.30)dt(b) If C is the graph of a smooth vector-valued function r(t) in 2-space or 3-space, wheres is an arc length parameter, then for every value of s the tangent vector to C haslengthdr∥ds∥ = 1 (4.31)(c) If C is the graph of a smooth vector-valued function r(t) in 2-space or 3-space, andif ‖dr/dt‖ = 1 for every value of t, then for any value of t 0 in the domain of r, theparameter s = t−t 0 is an arc length parameter that has its reference point at the pointon C where t = t 0 .The component forms of Formulas (4.30) and (4.31) are as follow:∥ √ (dx )ds ∥∥∥dt = dr2 dt∥ = +dt∥ √ (dx )ds ∥∥∥dt = dr2 dt∥ = +dt√ (dx ) dr2 ∥ds∥ = +ds√ (dx ) dr2 ∥ds∥ = +ds( ) 2 dy+ds( ) 2 dy+dt( ) 2 dy2-space (4.32)dt( ) 2 dz3-space (4.33)dt( ) 2 dy= 1 2-space (4.34)ds( ) 2 dz= 1 3-space (4.35)ds