What is the implicit equation of the plane through 3D points (1,0,0), (0, 1, 0), and (0, 0, 1)? What is the parametric equation? What is the normal vector to this plane?
What is the implicit equation of the plane through 3D points (1,0,0), (0, 1, 0), and (0, 0, 1)? What is the parametric equation? What is the normal
An implicit equation is a mathematical equation that defines a relationship between variables without expressing one variable explicitly in terms of the others. In other words, it describes a set of points that satisfy the equation are often used to represent complex or nonlinear relationships. For example, the equation of a circle in 2D, (x-h)^2+(y-k)^2 =r^2, is an implicit equation because it defines the points on the circle but doesn't directly solve for x or y.
A parametric equation is a set of equations that expresses each variable in terms of one or more independent parameters. It is commonly used to describe curves, surfaces, or other mathematical objects in a way that provides a parameterized representation of their points. Parametric equations are particularly useful for representing complex shapes and motions. For example, the parametric equations for a circle in 2D could be :
x = r * cos(format('truetype')%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22math12342d8a616837cd6d79017fe68%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%228.5%22%20y%3D%2216%22%3E%26%23x2205%3B%3C%2Ftext%3E%3C%2Fsvg%3E)
)
y = r* sin ( )
Here, (theta) is the parameter, and by varying theta, you can generate different points on the circle. Parametric equations provide a way to describe dynamic processes or curves that change over time or with other parameters.
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