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What information do we need to determine the equation of a plane in Cartesian xyz space
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We can find an equation for a plane in Cartesian xyz space if we know the direction of at least one vector that s perpendicular to the plane, and if we know the coordinates of at least one point in the plane We don t have to know the magnitude of the vector, but only its direction The point s coordinates don t have to tell us where the vector begins or ends; the point can be anywhere in the plane
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Imagine a plane that passes through a point whose coordinates are (x0,y0,z0) in Cartesian xyz space Also suppose that we ve found a vector ai + bj + ck that s normal (perpendicular) to the plane Based on this information, how can we write down an equation that represents the plane
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We can write the plane s equation in the standard form a(x x0) + b(y y0) + c(z z0) = 0 We can also write the equation as ax + by + cz + d = 0 where d is a constant that works out to d = ax0 by0 cz0
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What s the general equation for a sphere centered at the origin and having radius r in Cartesian xyz space
Answer 17-3
The equation can be written in the standard form x2 + y2 + z2 = r2
Question 17-4
What s the general equation for a sphere of radius r in Cartesian xyz space, centered at a point whose coordinates are (x0,y0,z0)
Answer 17-4
The equation can be written in the standard form (x x0)2 + (y y0)2 + (z z0)2 = r2
Question 17-5
Can a sphere have a negative radius in Cartesian xyz space
Answer 17-5
Normally, we define a sphere s radius as a positive real number Nevertheless, spheres with negative radii can exist in theory If we encounter a sphere whose radius happens to be defined
Review Questions and Answers
as a negative real number, then that sphere has the same equation as it would if we defined the radius as the absolute value of that number For all real numbers r, it s always true that r2 = |r|2, so the following two equations: (x x0)2 + (y y0)2 + (z z0)2 = r2 and (x x0)2 + (y y0)2 + (z z0)2 = |r |2 are equivalent, whether r is positive or negative
Question 17-6
What s the general equation for a distorted sphere in Cartesian xyz space
Answer 17-6
The equation can be written in the standard form (x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where (x0,y0,z0) are the coordinates of the center, a is the is the axial radius in the x direction, b is the axial radius in the y direction, and c is the axial radius in the z direction Normally, all three of the constants a, b, and c are positive reals
Question 17-7
There are three distinct classifications of distorted sphere What are they How can we tell, from the standard-form equation, which of these three types we have
Answer 17-7
We can have an oblate sphere, an ellipsoid, or an oblate ellipsoid We can tell which of these three types a particular standard-form equation represents by comparing the values of the axial radii a, b, and c We have an oblate sphere if and only if two of the positive real-number axial radii are equal, and the third is smaller In that case, one of the following is true: a<b=c b<a=c c<a=b We have an ellipsoid if and only if two of the positive real-number axial radii are equal, and the third is larger Then one of the following is true: a>b=c b>a=c c>a=b
Part Two 427
We have an oblate ellipsoid if and only if no two of the positive real-number axial radii are equal In that scenario, all of the following are true: a b b c a c
Question 17-8
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