© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 Unit 2, Lecture F Approximate Running Time - 15 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University Procedures: 1.Select “Slide Show” with the menu: Slide Show|View Show (F5 key), and hit “Enter” 2.You will hear “CHIMES” at the completion of the audio portion of each slide; hit the “Enter” key, or the “Page Down” key, or “Left Click” 3.You may exit the slide show at any time with the “Esc” key; and you may select and replay any slide, by navigating with the “Page Up/Down” keys, and then hitting “Shift+F5”.

© 2005 Baylor University Slide 2 Vector Equation of a Plane The Intercept Form The Linear Equation in 3-D is the Cartesian Equation of a Plane. Three points define a plane in 3-D Recall the Coplanar Vector Relationship The Vector Equation of a Plane or

© 2005 Baylor University Slide 3 The Cartesian equation of this plane: Equation of a Plane Example Given three points Substitute Write the equation for each coordinate The vector equation of a plane: Substitute or

© 2005 Baylor University Slide 4 Equation of a Plane - Matrix Method Given three points: Since A, B, C each are in the plane, their coordinates each satisfy the plane equation, using the Intercept Form: The Cartesian Equation of a Plane. or solve the three equations for the three unknown coefficients or gives: substituting:

© 2005 Baylor University Slide 5 Equation of a Plane - 4x4 Determinant Method or Three-Point Form Given three points: Create a 4x4 Determinant, using the Intercept Form: by Row Expansion: Note that the two matrix methods do not work for specialized planes, but the vector equation will always work or {

© 2005 Baylor University Slide 6 Three Methods for Equation of a Plane Solution of linear equations, using the Intercept Form: 4x4 Determinant Method, using the Intercept Form: Given three points, A, B, C in 3-D space Substitute into the vector equation of a plane: To obtain the Cartesian Equation of a Plane:

© 2005 Baylor University Slide 7 Picking a determines the place on the Line Parametric Vector Equation of a Line Given any two points, A, B in 3-D space The Parametric Vector Equation of a Line in 3-D:

© 2005 Baylor University Slide 8 Cartesian Equation of a Line also a form of the parametric vector equation of a line solve for : equating: In 3-D, the Cartesian equation of a line is the intersection of two plane equations. The Cartesian Equation of a Line in 3-D Expand the parametric vector equation to three equations

© 2005 Baylor University Slide 9 Equation of a Line Example Given two points: Cartesian form: parametric vector form:

© 2005 Baylor University Slide 10 This concludes Unit 2, Lecture F