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 \(\vec{a} \)

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:

\(\vec{AP} \)

From the figure given above it can be seen that,

\(\vec{AP} \)

Substituting this value in \(\vec{AP} \)

This equation represents the vector equation of a plane.

We will assume that P, Q and R points are regarded as x_{1}, y_{1}, z_{1} and x_{2}, y_{2}, z_{2} in respectively to change the equation into Cartesian system. A, B and C will be the assumed direction ratios. Thus,

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

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.

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