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Equations of a plane
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Vector and scalar product forms of a plane, cartesian equation of a plane
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Vector form
AHL 3.17
A plane in 3D space can be described by a vector equation involving a fixed point and two direction vectors lying in the plane. Planes are often denoted by Π (capital pi).
If the position vector of the fixed point is a, and two non-parallel direction vectors in the plane are b and c, then the plane is represented by:
Π:r=a+λb+μc📖
Here, λ and μ are parameters that can take any real values, allowing r to move freely across the entire surface of the plane.
Scalar product form
AHL 3.17
The scalar product form of a plane uses a vector perpendicular ("normal") to the plane and one known point in the plane. If a point in the plane has position vector a and n is a normal vector, then any other point r lies in the plane if:
Π:r⋅n=a⋅n📖
This equation expresses the idea that the vector from the known point to any other point in the plane is always perpendicular to n.
Cartesian equation of a plane
AHL 3.17
The Cartesian equation of a plane with a normal vector n and containing a point with position vector a is
n1x+n2y+n3z=d📖
where n=⎝⎛n1n2n3⎠⎞,d=a⋅n.
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