Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 51(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+8t, y = 4−3t, z = 5+tTwo 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 .........Vector Equations of LinesWe will now show how vector notation can be used to express the parametric equations ofa line. Because two vectors are equal if and only if their components are equal, (3.22) and(3.23) can be written in vector form as〈x,y〉 = 〈x 0 +at,y 0 +bt〉〈x,y,z〉 = 〈x 0 +at,y 0 +bt,z 0 +ct〉