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This tells us that (2,3,4) is a point on the line We can also see that a=4 b=3 c=2 so the line s direction numbers are (4,3,2) We can write down a standard-form direction vector m from these numbers as m = 4i + 3j + 2k
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From algebra, we remember that a quadratic equation in a variable x can always be written in the form a1x2 + a2x + a3 = 0 where a1, a2, and a3 are real-number constants called the coefficients, and a1 0 If we replace the 0 on the right-hand side of this equation by another variable and then transpose the sides, we get an expression for a quadratic function For example, 4x2 + 2x + 1 = 0 is a quadratic equation in x, but y = 4x2 + 2x + 1 is a quadratic function in which the independent variable is x and the dependent variable is y If we give our function a name (f, for example), then we can denote it as f (x) = 4x2 + 2x + 1 When we graph a quadratic function in Cartesian two-space, we always get a parabola that s fairly easy to graph, because there s only one plane to worry about (the xy plane, if our independent variable is x and our dependent variable is y) In xyz space, the situation is more complicated, because we have an extra variable There are infinitely many different planes in which a parabola can lie, as well as infinitely many different shapes and orientations for a parabola in any particular plane Let s look at a few simple cases
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Hold x constant Imagine a parameter t that s allowed to wander all over the set of real numbers Also imagine a generalized quadratic function f of this parameter, such that
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f (t) = a1t2 + a2t + a3
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(c, 0, 0)
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Parabola in plane x=c
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Parabola in a plane where x is held to a constant value c The plane is perpendicular to the x axis, and intersects that axis at the point (c,0,0)
where a1, a2, and a3 are real-number coefficients Let s go into Cartesian xyz space and restrict ourselves to a single plane in which the value of x is some real-number constant c This plane is parallel to the yz plane, and it intersects the x axis at the point (c,0,0) Consider a parabola in the plane x = c whose axis is parallel to the y axis, as shown in Fig 18-3 (The axis of a parabola is a straight line in the same plane as the parabola, and on either side of which the parabola is symmetrical) In this situation, the value of z tracks right along with the value of t, while the variable y follows f (t) Therefore x=c y = f (t) = a1t2 + a2t + a3 z=t The above set of equations is a parametric description of our parabola If we want to describe a parabola in the plane x = c whose axis is parallel the z axis instead of the y axis, then y follows t while z follows f (t), and we have x=c y=t z = f (t) = a1t2 + a2t + a3
Lines and Curves in Three-Space
Hold y constant Now suppose that we restrict our movements to a plane in which the value of y is always equal to a constant c The equation of the plane is y = c It s parallel to the xz plane, and it intersects the y axis at (0,c,0) Imagine a parabola in this plane whose axis is parallel to the z axis, as shown in Fig 18-4 In this situation, x follows t while z follows f (t), and the curve can be described as
x=t y=c z = f (t) = a1t2 + a2t + a3 To describe a parabola in the plane y = c whose axis is parallel the x axis, we can let z follow t and let x follow f (t), getting the system x = f (t) = a1t2 + a2t + a3 y=c z=t
Hold z constant Finally, let s confine our attention to a single plane in which the value of z is some real-number constant c The plane z = c is parallel to the xy plane, and it intersects the z axis at (0,0,c)