Difference between revisions of "Plane"
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| − | A plane is a flat surface | + | A '''plane''' is a flat surface with limitless area. |
| + | It takes only three [[point]]s to determine a plane, but they must be non-[[collinear]], i.e., they can't all be on the same line. | ||
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| + | Mathematically, a plane takes the form Ax + By + Cz = Ax<sub>0</sub> + By<sub>0</sub> + Cz<sub>0</sub>, where the point (x<sub>0</sub>, y<sub>0</sub>, z<sub>0</sub>) is in the plane, and the three-dimensional [[vector]] <A, B, C> is normal (also orthogonal or [[perpendicular]]) to the plane. | ||
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| + | When the principle of a plane is extended into higher dimensions, it is known as a [[hyperplane]]. | ||
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| + | == How to find the equation of a plane from three points == | ||
| + | Consider points A(x<sub>A</sub>, y<sub>A</sub>, z<sub>A</sub>), B(x<sub>B</sub>, y<sub>B</sub>, z<sub>B</sub>), and C(x<sub>C</sub>, y<sub>C</sub>, z<sub>C</sub>). How do you find the equation of the plane defined by those three points? | ||
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| + | First, you must find the vector normal to the plane. This is found by calculating the [[cross product]] of two vectors in the plane. | ||
| + | <br>vector AB = <(x<sub>B</sub> - x<sub>A</sub>), (y<sub>B</sub> - y<sub>A</sub>), (z<sub>B</sub> - z<sub>A</sub>)> | ||
| + | <br>vector BC = <(x<sub>C</sub> - x<sub>B</sub>), (y<sub>C</sub> - y<sub>B</sub>), (z<sub>C</sub> - z<sub>B</sub>)> | ||
| + | <br>AB cross BC = <D, E, F> | ||
| + | <br>(D, E, and F stand for the numbers found when the cross product is calculated.) | ||
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| + | The equation of the plane is then D(x - x<sub>A</sub>) + E(y - y<sub>A</sub>) + F(z - z<sub>A</sub>) = 0 | ||
| + | <br>(The coordinates of points B and C could have been used instead.) | ||
| + | <br> | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
Latest revision as of 17:36, July 13, 2016
A plane is a flat surface with limitless area. It takes only three points to determine a plane, but they must be non-collinear, i.e., they can't all be on the same line.
Mathematically, a plane takes the form Ax + By + Cz = Ax0 + By0 + Cz0, where the point (x0, y0, z0) is in the plane, and the three-dimensional vector <A, B, C> is normal (also orthogonal or perpendicular) to the plane.
When the principle of a plane is extended into higher dimensions, it is known as a hyperplane.
How to find the equation of a plane from three points
Consider points A(xA, yA, zA), B(xB, yB, zB), and C(xC, yC, zC). How do you find the equation of the plane defined by those three points?
First, you must find the vector normal to the plane. This is found by calculating the cross product of two vectors in the plane.
vector AB = <(xB - xA), (yB - yA), (zB - zA)>
vector BC = <(xC - xB), (yC - yB), (zC - zB)>
AB cross BC = <D, E, F>
(D, E, and F stand for the numbers found when the cross product is calculated.)
The equation of the plane is then D(x - xA) + E(y - yA) + F(z - zA) = 0
(The coordinates of points B and C could have been used instead.)