Analytic Geometry – Plane

Jumat, 05 Juni 2020

Plane

 

Point coordinater: x, y, z, x₀, y₀, z₀, x₁, y₁, z₁, …

Real numbers: A, B, C, D, A₁, A₂, a, b, c, a₁, a₂, λ, p, t, …

Normal vectors: 

Direction cosines: cos α, cos β, cos γ

Distance from point to plane: d

 

• General Equation of a Plane

   Ax + By + Cz + D = 0

 

• Normal Vector to a Plane

   The vector  (A, B, C) is normaL to the plane

 

Figure

 

• Particular Cases of the Equation of a Plane

   Ax + By + Cz + D = 0

 

   If A = 0, the plane is parallel to the x-axis.

   If B = 0, the plane is parallel to the y-axis.

   If C = 0, the plane is parallel to the z-axis.

   If A = 0, the plane lies on the origin.

 

   If A = B, the plane is parallel to the xy-plane.

   If B = C, the plane is parallel to the yz-plane.

   If A = C, the plane is parallel to the xz-plane.

 

 

• Point Direction Form

   A(x – x₀) + B(y – y₀) + C(z – z₀) = 0,

   Where the point P(x₀, y₀, z₀) lies in the plane, and the vector (A, B, C) 

   normaL to the  plane

 

Figure

 

 

• Intercept Form

     

 

Figure

 

• Three Point Form

      

 

Figure

 

 

• Normal Form

   x cos α + y cos β + z cos γ – p = 0,

   where p is the perpendicular distance from the origin to the plane, and 

   cos α, cos β, cos γ are the direction cosine of any ine normal to the plane.

 

Figure

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Labels: Mathematician

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