Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 50Theorem 3.14(a) The line in 2-space that passes through the point P 0 (x 0 ,y 0 ) and is parallel to thenonzero vector v = 〈a,b〉 = ai+bj has parametric equationsx = x 0 +at, y = y 0 +bt (3.22)(b) The line in 3-space that passes through the point P 0 (x 0 ,y 0 ,z 0 ) and is parallel to thenonzero vector v = 〈a,b,c〉 = ai+bj+ck has parametric equationsx = x 0 +at, y = y 0 +bt, z = z 0 +ct (3.23)Example 3.27 Find parametric equations of the line passing through the point (1,5,2) andparallel to the vector v = 〈4,3,7〉. Also, determine where the line intersects the yz-plane.Solution .........Example 3.28 Findparametricequationsof the lineLpassingthrough the pointP(1,2,−1)and Q(5,−3,4).Solution .........Example 3.29 Let L 1 and L 2 be the lines(a) Are the lines parallel?(b) Do the lines intersect?Solution .........L 1 : x = 1+4t, y = 5−4t, z = −1+5tL 2 : x = 2+8s, y = 4−3s, z = 5+sTwo lines in 3-space that are not parallel and do not intersect are called skew lines.Line SegmentsSometimes one is not interested in an entire line, but rather some segment of a line. Parametricequations of a line segment can be obtained by finding parametric equation for theentire line, then restricting the parameter appropriately so that only the desired segment isgenerated.Example 3.30 Find parametric equations describing the line segment joining the pointsP(1,2,−1) and Q(5,−3,4).Solution .........