Finding formulas can be frustrating. Believe us, we have been there. Having a handful of formulas written on a single page can help us and increases productivity. That is why this resource is dedicated to all plane formulas that you require to solve plane problems.

Vectorial Equation of the Plane

The point P belongs to the plane \pi if the vector \vec { PX } is coplanar with the vectors \vec { u } and \vec { v }.

\vec { PX } = \lambda \vec { u } + \mu \vec { v }

However, the vector \vec { PX } is also formed by two vectors from x,y, and z coordinates. This vector is formed from the vector \vec { OP } and \vec { OX }.

\vec { OP } - \vec { OX } = \lambda \vec { u } + \mu \vec { v }

\vec { OP } = \vec { OX } + \lambda \vec { u } + \mu \vec { v }

Let's check the coordinate form.

(x - { x }_{ 0 }, y - { y }_{ 0 }, z - { z }_{ 0 }) = \lambda ({ u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 }) + \mu ({ v }_{ 1 }, { v }_{ 2 }, { v }_{ 3 })

(x, y, z) = ({ x }_{ 0 }, { y }_{ 0 }, { z }_{ 0 }) + \lambda ({ u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 }) + \mu ({ v }_{ 1 }, { v }_{ 2 }, { v }_{ 3 })

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Parametric Equations of the Plane

\left\{\begin{matrix} x = { x }_{ 0 } + { u }_{ 1 } \lambha + { v }_{ 1 } \mu \\ y = { y }_{ 0 } + { u }_{ 2 } \lambha + { v }_{ 2 } \mu \\ z = { z }_{ 0 } + { u }_{ 3 } \lambha + { v }_{ 3 } \mu \end{matrix}\right

Cartesian Equation of the Plane

\begin{vmatrix} x - { x }_{ 0 } & { u }_{ 1 } & { v }_{ 1 } \\ y - { y }_{ 0 } & { u }_{ 2 } & { v }_{ 2 } \\ z - { z }_{ 0 } & { u }_{ 3 } & { v }_{ 3 } \end{vmatrix} = 0

Ax + By + Cz + D = 0

Intercept Form

A(a, 0, 0), B(0, b, 0) and C(0, 0, c).

\frac { x }{ a } + \frac { y }{ b } + \frac { z }{ c } = 1

a = \frac { -D }{ A } \qquad b = \frac { -D }{ B } \qquad c = \frac { -D }{ C }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.