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Transcript
MA242.003 •Day 9 – January 17, 2013 •Review: Equations of lines, Section 9.5 •Section 9.5 –Planes Equations of PLANES in space. Equations of PLANES in space. Different ways to specify a plane: Equations of PLANES in space. Different ways to specify a plane: 1. Give three non-co-linear points. Equations of PLANES in space. Different ways to specify a plane: 1. Give three non-co-linear points. Equations of PLANES in space. Different ways to specify a plane: 2. Give two non-parallel intersecting lines. Equations of PLANES in space. Different ways to specify a plane: 2. Give two non-parallel intersecting lines. Equations of PLANES in space. Different ways to specify a plane: 3. Specify a point and a normal vector Given: Equations of PLANES in space. Equations of PLANES in space. Example: Find an equation for the plane containing the point (1,-5,2) with normal vector <-3,7,5> Example: Find an equation for the plane containing the the points P=(1,-5,2), Q=(-3,8,2) and R=(0,-1,4) REMARK: How equations of planes occur in problems REMARK: How equations of planes occur in problems The Geometry of Lines and Planes • For us, a LINE in space is a The Geometry of Lines and Planes • For us, a LINE in space is a Point and a direction vector v = <a,b,c> The Geometry of Lines and Planes • For us, a Plane in space is a The Geometry of Lines and Planes • For us, a Plane in space is a Point on the plane And a normal vector n = <a,b,c> Two lines are parallel Two lines are parallel when Two lines are parallel their direction vectors are parallel when Two lines are perpendicular Two lines are perpendicular when Two lines are perpendicular their direction vectors are orthogonal when Two planes are parallel Two planes are parallel when Two planes are parallel Their normal vectors are parallel when Two planes are perpendicular Two planes are perpendicular when Two planes are perpendicular Their normal vectors are orthogonal when A line is parallel to a plane A line is parallel to a plane when A line is parallel to a plane when the direction vector v for the line is orthogonal to the normal vector n for the plane A line is perpendicular to a plane A line is perpendicular to a plane when A line is perpendicular to a plane when the direction vector v for the line is parallel to the normal vector n for the plane Example Problems Example Problems Example Problems Example Problems