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Analytic Geometry2D

Solid Geometry - Theory & Concepts - Analytic Geometry Plane Three Points3 D

Three-dimensional coordinate systems, distance, direction cosines, equations of lines and planes in space.

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Plane Defined by Three Points

x₁2.0
y₁1.0
z₁3.0
x₂-1.0
y₂3.0
z₂1.0
x₃1.0
y₃-2.0
z₃-1.0
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1. Direction Vectors
v⃗=P1P2→=⟨−3.0,2.0,−2.0⟩\vec{v} = \overrightarrow{P_1P_2} = \langle -3.0, 2.0, -2.0 \ranglew⃗=P1P3→=⟨−1.0,−3.0,−4.0⟩\vec{w} = \overrightarrow{P_1P_3} = \langle -1.0, -3.0, -4.0 \rangle
2. Normal Vector
n⃗=v⃗×w⃗=⟨A,B,C⟩\vec{n} = \vec{v} \times \vec{w} = \langle A, B, C \ranglen⃗=⟨−14.0,−10.0,11.0⟩\vec{n} = \langle -14.0, -10.0, 11.0 \rangle
3. Plane Equation
A(x−x1)+B(y−y1)+C(z−z1)=0A(x - x_1) + B(y - y_1) + C(z - z_1) = 0
gives:
(−14.0)x+(−10.0)y+(11.0)z+(5.0)=0(-14.0)x + (-10.0)y + (11.0)z + (5.0) = 0

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