Chapter 1 Topics in Analytic Geometry

•MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 65Example 4.5 Describe the graph of the vector-valued functionSolution .........r(t) = 〈cost,sint,t〉 = costi+sintj+tkIf a parametric curve is viewed as the graph of a vector-valued function, then we canalso imagine the graph to be traced by the tip of the moving vector. For example, if thecurve C in 3-space is the graph ofr(t) = x(t)i+y(t)j+z(t)kand if we position r(t) so its initial point is at the origin, then its terminal point will fallon the curve C.z(x(t),y(t),z(t))r(t)CyxThus, when r(t) is positioned with its initial point at the origin, its terminal point will traceout the curve C as the parameter t varies, in which case we call r(t) the radius vector orthe position vector for C.Example 4.6 Sketch the graph and a radius vector of(a) r(t) = costi+sintj, 0 ≤ t ≤ 2π(b) r(t) = costi+sintj+2k, 0 ≤ t ≤ 2πSolution .........Vector Form of a Line SegmentRecall that if r 0 is a vector in 2-space or 3-space with its initial points at the origin, thenthe line that passes through the terminal point of r 0 and is parallel to the vector v can beexpressed in vector form asr = r 0 +tvIn particular, if r 0 and r 1 are vectors in 2-space or 3-space with their initial points atthe origin, then the line that passes through the terminal points of these vectors can beexpressed in vector form asr = r 0 +t(r 1 −r 0 ) or r = (1−t)r 0 +tr 1 (4.5)