Equation of A Plane - 3D Geometry

A Vector is a physical quantity that with it’s magnitude also has a direction attached to it. With reference to a origin the position vector basically denotes the location or position (in a 3D Cartesian system) of a point. The Cartesian equation of a plan in 3 Dimensional space and vectors are explored in this post.

Equation of a plane

There are infinite planes that lie perpendicular to a specific vector. But only one unique plane exists to a specific point which remains perpendicular to the point while going through it.

Let us consider a plane passing through a given point A having position vector   \(\vec{a} \) and perpendicular to the vector  \(\vec{N} \) . Let us consider a point P(x, y, z) lying on this plane and its position vector is given by  \(\vec{r} \) as shown in the figure given below.

equation of a plane

Position vector simply denotes the position or location of a point in the three dimensional Cartesian system with respect to a reference origin.

For point P to lie on the given plane it must satisfy the following condition:

\(\vec{AP} \)  is perpendicular to \(\vec{N} \)  , i.e. \(\vec{AP} \).\(\vec{N} \) =0

From the figure given above it can be seen that,

\(\vec{AP} \) = ( \(\vec{r} \) – \(\vec{a} \))

Substituting this value in  \(\vec{AP} \)\(\vec{N} \) =0, we have (\(\vec{r} \) – \(\vec{a} \)). \(\vec{N} \) =0

This equation represents the vector equation of a plane.

We will assume that P, Q and R points are regarded as x1, y1, z1 and x2, y2, z2 in respectively to change the equation into Cartesian system. A, B and C will be the assumed direction ratios. Thus,

equation of a plane

Substituting these values in the vector equation of a plane, we have

r-a vec

This gives us the Cartesian equation of a plane. To learn more about equation of a plane in three dimensions and three dimensional geometry download Byju’s- The Learning App.

Practise This Question

If 3x+7=13, the x is