Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 54If preferred, we can express this vector equation in terms of components as〈a,b,c〉·〈x−x 0 ,y −y 0 ,z −z 0 〉 = 0 (3.30)from which we obtaina(x−x 0 )+b(y −y 0 )+c(z −z 0 ) = 0 (3.31)This is called the point-normal form of the equation of a plane. Formula (3.29) and(3.30) are vector version of this formula.Example 3.33 Find an equation of the plane passing through the point (1,2,3) and perpendicularto the vector n = 〈4,5,6〉.Solution .........Observe that if we multiply out the terms in (3.31) and simplify, we obtain an equationof the formax+by +cz +d = 0For example, equation in Example 3.33 can be rewritten as4x+5y +6z −32 = 0Theorem 3.15 If a, b, c, and d are constants, and a, b, and c are not all zero, then thegraph of the equationax+by +cz +d = 0 (3.32)is a plane that has the vector n = 〈a,b,c〉 as a normal.Equation (3.32) is called the general form of the equation of a plane.Example 3.34 Determine whether the planeare parallel.Solution .........3x−4y +5z = 0 and −6x+8y −10z −4 = 0Example 3.35 Find an equation of the plane through the points P 1 (1,2,2), P 2 (2,−1,4),and P 3 (3,5,−2).Solution .........Example 3.36 Determine whether the lineis parallel to the plane x−3y +5z = 12.Solution .........x = 3+8t, y = 4+5t, z = −3−tExample 3.37 Find the intersection of the line and the plane in Example 3.36.Solution .........Example 3.38 Find an equation for the plane through the point (1,4,−5) and parallel tothe plane defined by 2x−5y +7z = 12.Solution .........