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What s the general equation for a hyperboloid of one sheet in Cartesian xyz space
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The equation can be written in one of the following standard forms: (x x0)2/a2 + (y y0)2/b2 (z z0)2/c2 = 1 (x x0)2/a2 (y y0)2/b2 + (z z0)2/c2 = 1 (x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where (x0,y0,z0) are the coordinates of the center, the constants a, b, and c are positive real numbers that define the object s general shape, and the locations of the signs (plus and minus) define the object s orientation with respect to the coordinate axes
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What s the general equation for a hyperboloid of two sheets in Cartesian xyz space
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The equation can be written in one of the following standard forms: (x x0)2/a2 + (y y0)2/b2 (z z0)2/c2 = 1 (x x0)2/a2 (y y0)2/b2 (z z0)2/c2 = 1 (x x0)2/a2 (y y0)2/b2 + (z z0)2/c2 = 1 where (x0,y0,z0) are the coordinates of the center, the constants a, b, and c are positive real numbers that define the object s general shape, and the locations of the signs (plus and minus) define the object s orientation with respect to the coordinate axes
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What s the general equation for an elliptic cone in Cartesian xyz space
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The equation can be written in one of the following standard forms: (x x0)2/a2 + (y y0)2/b2 (z z0)2/c2 = 0 (x x0)2/a2 (y y0)2/b2 + (z z0)2/c2 = 0 (x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 0
Review Questions and Answers
where (x0,y0,z0) are the coordinates of the point where the apexes of the two halves of the double cone meet, the constants a, b, and c are positive real numbers that define the object s general shape, and the locations of the signs (plus and minus) define the object s orientation with respect to the coordinate axes Don t get confused by the similarity between these equations and those for hyperboloids of one sheet The only difference is that the net values are all equal to 1 for the hyperboloids, and all equal to 0 for the cones
18
Question 18-1
What s the general symmetric equation for a straight line in Cartesian xyz space
Answer 18-1
Imagine that (x0,y0,z0) are the coordinates of a specific point Suppose that a, b, and c are nonzero real-number constants The general symmetric equation of a straight line passing through (x0,y0,z0) is (x x0)/a = (y y0)/b = (z z0)/c The constants a, b, and c are called direction numbers When considered all together as an ordered pair (a,b,c), these numbers define the direction or orientation of the line with respect to the coordinate axes
Question 18-2
What are the general parametric equations for a straight line in Cartesian xyz space
Answer 18-2
Let (x0,y0,z0) be the coordinates of a specific point, and suppose that a, b, and c are nonzero real-number constants The general parametric equations for a straight line passing through (x0,y0,z0) are x = x0 + at y = y0 + bt z = z0 + ct where the parameter t can range over the entire set of real numbers As with the symmetric form, the constants a, b, and c are direction numbers that tell us how the line is orientated relative to the coordinate axes
Question 18-3
What is meant by the expression preferred direction numbers when describing the orientation of a straight line in Cartesian xyz space
Answer 18-3
For any line in Cartesian xyz space, there are infinitely many ordered triples that can define its orientation with respect to the coordinate axes For example, if a line has the direction numbers (a,b,c), then we can multiply all three entries by a real number other than 0 or 1, and we ll get
Part Two 429
another valid ordered triple of direction numbers for that same line For the sake of simplicity and elegance, mathematicians usually reduce the direction numbers so that their only common divisor is 1, and so that at most one of them is negative Doing this produces a unique set of direction numbers in lowest terms, and these are the preferred direction numbers for the line
Question 18-4
What are the generalized parametric equations for a parabola in Cartesian xyz space where the value of x is constant, and the curve s axis is parallel to either the y axis or the z axis
Answer 18-4
If x is constant and the axis of the parabola is parallel to the y axis, then the curve s parametric equations are x=c y = a1t2 + a2t + a3 z=t where a1, a2, and a3 are real-number coefficients, c is the real-number constant value to which x is held, and t is a parameter that can range over the set of all real numbers If x is constant and the curve s axis is parallel to the z axis, then the parametric equations are x=c y=t z = a1t2 + a2t + a3 In either case, the parabola lies in a plane parallel to the yz plane
Question 18-5
What are the generalized parametric equations for a parabola in Cartesian xyz space where the value of y is constant, and the curve s axis is parallel to either the x axis or the z axis
Answer 18-5
If y is constant and the axis of the parabola is parallel to the x axis, then the parametric equations are x = a1t2 + a2t + a3 y=c z=t where a1, a2, and a3 are real-number coefficients, c is the real-number constant value to which y is held, and t is a parameter that can range over the set of all real numbers If y is constant and the curve s axis is parallel to the z axis, then the parametric equations are x=t y=c z = a1t2 + a2t + a3 In either case, the parabola lies in a plane parallel to the xz plane
Review Questions and Answers Question 18-6
What are the generalized parametric equations for a parabola in Cartesian xyz space where the value of z is constant, and the curve s axis is parallel to either the x axis or the y axis
Answer 18-6
If z is constant and the axis of the parabola is parallel to the x axis, then the parametric equations are x = a1t2 + a2t + a3 y=t z=c where a1, a2, and a3 are real-number coefficients, c is the real-number constant value to which z is held, and t is a parameter that can range over the set of all real numbers If z is constant and the curve s axis is parallel to the y axis, then the parametric equations are x=t y = a1t2 + a2t + a3 z=c In either case, the parabola lies in a plane parallel to the xy plane
Question 18-7
What are the generalized parametric equations for a circle in Cartesian xyz space where the value of x is constant so the circle lies in a plane parallel to the yz plane, and the center of the circle lies on the x axis
Answer 18-7
The parametric equations are x=c y = r cos t z = r sin t where r is the radius of the circle, c is the real-number constant value to which x is held, and t is a parameter that varies continuously over a real-number interval at least 2p units wide
Question 18-8
What are the generalized parametric equations for a circle in Cartesian xyz space where the value of y is constant so the circle lies in a plane parallel to the xz plane, and the center of the circle lies on the y axis
Part Two 431 Answer 18-8
The parametric equations are x = r cos t y=c z = r sin t where r is the radius of the circle, c is the real-number constant value to which y is held, and t is a parameter that varies continuously over a real-number interval at least 2p units wide
Question 18-9
What are the generalized parametric equations for a circle in Cartesian xyz space where the value of z is constant so the circle lies in a plane parallel to the xy plane, and the center of the circle lies on the z axis
Answer 18-9
The parametric equations are x = r cos t y = r sin t z=c where r is the radius of the circle, c is the real-number constant value to which z is held, and t is a parameter that varies continuously over a real-number interval at least 2p units wide
Question 18-10
Imagine a circle in Cartesian xyz space whose sets of parametric equations have one of the forms described in Answer 18-7 through 18-9 Consider the variable that s held constant Suppose that, instead of insisting that it always keep the same value, we allow that variable to follow a constant multiple of the parameter t What sort of curve will we get under these conditions
Answer 18-10
We ll get a circular helix centered on the axis for whichever variable follows the constant multiple of t
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