Chapters

Vectorial Equation of the Plane
Parametric Equations of the Plane
Cartesian Equation of the Plane
Intercept Form

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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.

Plane Formulas

Vectorial Equation of the Plane

Parametric Equations of the Plane

Cartesian Equation of the Plane

Intercept Form

Theory

Irregular Polygons

Regular Pentagons

Points, Lines and Planes

Quadrilaterals and Regular Polygons

Rhombus

Triangle

Diagonals of a Polygon

Concentric Circles

Area and Perimeter of a Triangle

Circular Sectors

Regular Polygons

Similar Triangles

Angles of a Polygon

Parallelograms

Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

Trapezoids

Circumference

Equilateral Triangles

Circles

Quadrilaterals

Circular Segments

Angle Bisectors

Coplanar

Angles of the Triangle

Angles

Apothems

Intersection of Three Planes

Height of a Polygon

Polygon

Plane Equation

Line Segments

Points

Medians of a Triangle

Star Polygons

Inscribed Polygons

Sheaf of Planes

Lune of Hippocrates

Area and Perimeter of Polygons

Circumscribed Polygons

Polygons

Rectangle

Right Triangle

Lines

Regular Hexagons

Plane

Triangles

Rhomboid

Sides of a Polygon

Perpendicular Bisectors

Intersection of Two Planes

Squares

Formulas

Circle Formulas

Geometry Formulas

Triangle Formulas

Area and Perimeter Formulas

Area Formulas

Perimeter Formulas

Exercises

Rectangle Problems

Trapezoid Problems

Pythagorean Theorem

Pythagorean Theorem Worksheet

Area Word Problems

Circle Word Problems

Pythagorean Theorem Word Problems

Plane Problems

Square Problems

Circle Worksheet

Triangle Problems

Area Worksheet

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