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8 Consider the pair of parametric equations x = a csc t and y = b cot t where a and b are nonzero real-number constants Find an equation for this relation in terms of x and y only, without the parameter t What sort of curve should expect to get if we graph this relation in the Cartesian xy plane 9 Express the relation x = sin (cos y) as a pair of parametric equations 10 Manipulate the equation stated in Problem 9 so that y appears all by itself on the left-hand side of the equals sign, and operations involving x appear on the right-hand side Then manipulate your answer to Problem 9 to get the same equation
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CHAPTER
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Three-space can contain an infinite variety of surfaces, all of which can be defined as equations in terms of three variables In this chapter, we ll examine a few basic surfaces and their equations in Cartesian three-space
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An intuitive way to express the equation for a plane in Cartesian xyz space is to define the direction of a vector normal (perpendicular) to the plane, and then to identify the coordinates of a point in the plane We don t have to know the magnitude of the vector, and the point in the plane doesn t have to be the one where the vector originates
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General equation of plane Figure 17-1 shows a plane W in Cartesian three-space, a point P = (x0,y0,z0) in the plane W, and a vector (a,b,c) = ai + bj + ck that s normal to plane W The vector (a,b,c) originates at a point Q that differs from P, and which is also located away from the coordinate origin The values x = a, y = b, and z = c for the vector are nevertheless based on the vector s standard form, as if it originated at (0,0,0) The point and the vector give us enough information to uniquely define the plane and write its equation in standard form as
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a(x x0) + b(y y0) + c(z z0) = 0 This equation can also be written as ax + by + cz + d = 0 where d is a stand-alone constant With a little algebra, we can work out its value in terms of the other constants and coefficients as d = ax0 by0 cz0
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Vector (a, b, c) normal to W at point Q Point P (x0, y0 , z0) in plane W
Point Q in plane W
Plane W
Figure 17-1 A plane W can be uniquely defined on the basis of a point
P in the plane and a vector (a,b,c) normal to the plane
Plotting a plane When we want to construct a plane in Cartesian xyz space based on its equation, we can do it by figuring out the coordinates of points where the plane crosses each of the three coordinate axes These points are the x-intercept, the y-intercept, and the z-intercept When we plot these intercept points on the axes, we can envision the position and orientation of the plane There s a potential hangup with this scheme for plane-graphing Not all planes cross all three axes in Cartesian xyz-space If a plane is parallel to one of the axes, then it does not cross that axis, although must cross at least one of the other two If a plane is parallel to the plane formed by two coordinate axes, then that plane crosses only the axis with respect to which it is not parallel An example Suppose that a plane contains the point (3, 6,2), and the standard form of a vector normal to the plane is 4i + 3j + 2k Let s find the plane s equation in the standard form given above To begin, we know that the vector
4i + 3j + 2k is equivalent to the ordered triple (a,b,c) = (4,3,2)
Surfaces in Three-Space
We ve been told that (x0,y0,z0) = (3, 6,2) and that this point lies in the plane The general formula for the plane is a(x x0) + b(y y0) + c(z z0) = 0 Plugging in the known values for a, b, c, x0, y0, and z0, we get 4(x 3) + 3[y ( 6)] + 2(z 2) = 0 which simplifies to 4x + 3y + 2z + 2 = 0