Chapter 1 Topics in Analytic Geometry

••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 52or, equivalently, as〈x,y〉 = 〈x 0 ,y 0 〉+t〈a,b〉 (3.24)〈x,y,z〉 = 〈x 0 ,y 0 ,z 0 〉+t〈a,b,c〉 (3.25)For the equation in 2-space we define the vectors r, r 0 and v asand for the equation in 3-space we define them asr = 〈x,y〉, r 0 = 〈x 0 ,y 0 〉, v = 〈a,b〉 (3.26)r = 〈x,y,z〉, r 0 = 〈x 0 ,y 0 ,z 0 〉, v = 〈a,b,c〉 (3.27)Substituting (3.26) and (3.27) in (3.24) and (3.25), respectively, yields the equationr = r 0 +tv (3.28)in both case. We call this the vector equation of a line in 2-space or 3-space. In thisequation, v is a nonzero vector parallel to the line, and r 0 is a vector whose componentsare the coordinates of a point on the line.ytvLr 0P 0vrtvxExample 3.31 The equation〈x,y,z〉 = 〈−1,0,2〉+t〈1,5,−4〉is of form (3.28) withr 0 = 〈−1,0,2〉, v = 〈1,5,−4〉Thus, the equation represents the line in 3-space that passes through the point (−1,0,2) andis parallel to the vector 〈1,5,−4〉.✠Example 3.32 Find an equation of the line in 3-space that passes through the pointsP 1 (2,4,−1) and P 2 (5,0,7).Solution .........