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Ch12Notes - Chapter 12 Functions of Several Variables...

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Chapter 12: Functions of Several Variables Section 12.1 (PLANES and SURFACES) Equation of a plane in space in point-normal form and standard form Must know a point on the plane and a normal vector to the plane Point-Normal Form: Standard Form: (the numbers a, b, and c are called attitude numbers) Find the equation of a plane given: 1) 3 points in the plane 2) A point in the plane and an orthogonal line to the plane 3) A point in the plane and a parallel plane to the plane 4) Two intersecting lines in the plane Find the line of intersection of two planes Since the line is in both planes any point on the line satisfies both equations for the planes. We can set up 2 equations with 3 unknowns (x , y , and z). Let z = 0 and solve the system. We have a point on the line. Now we need the direction of the line. We know the direction of the line is perpendicular to both normal vectors of the two planes. So take the cross product of the normal vectors to get the direction vector of the line. Now we have enough information to write the equation of the line. Find the distance from: 1) A point to a plane (also finding the equation of a sphere given center and tangent to a plane) where is the normal vector to the plane, Q is a point in the plane and P is the point we are trying to find the distance from. Derive the above formula by drawing a picture that shows the relationships: P Q
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2) Between two parallel planes where is the normal vector to the planes, Q is a point in the plane and P is a point in the other plane. Derive the above formula by drawing a picture that shows the relationships: Find the angle between two planes: This can be accomplished by finding the angle between the two normal vectors. Normal Vector for Plane 2 Normal vector for Plane 1 Plane 1 Plane 2 A + = 90 o Sketching a Plane If the equation of the plane has all three variables then find the three intersection points with the coordinate axes. Connect the three points to get a triangle which represents part of the plane. P Q
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Surfaces in Space Sketching cylinders and the 7 surfaces using trace curves Identifying the type of surface from the equation Notes for 12.1 General Second Degree Equation in TWO variables You have already studied these (quadric) curves but we referred to them as Conic Sections Circles , Ellipses , Parabolas , Hyperbolas and the Degenerate Conics: Point , Line , Pair of Intersecting Lines , Parallel Lines , No Graph A conic section may more formally be defined as the locus of a point that moves in the plane of a fixed point called the focus and a fixed line called the conic section directrix (with not on ) such that the ratio of the distance of from to its distance from is a constant called the eccentricity . If , the conic is a circle , if , the conic is an ellipse , if , the conic is a parabola , and if , it is a hyperbola .
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