Perpendicular Distance of a Point from a Plane Formula

Perpendicular distance of a point to a plane is defined as the shortest distance covered from one point to a plane. It is known as the length of the perpendicular which is drawn from that one point to touch the plane. Perpendicular distance of a point from a plane is calculated by using two forms namely vector form and cartesian form. These are the two forms which will help us in measuring the shortest distance between the point and the plane. 

Key Takeaways: Perpendicular, Point on a Plane, Vector Form, Cartesian Form.

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Perpendicular Distance of a Point from a Plane

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The shortest distance covered from one point to a plane is known to be along the line which is perpendicular to the plane, or in other words, is known as the perpendicular distance of that one point from the plane. Hence, if we take a normal vector, suppose for example n, to the plane given, a line which runs parallel to n vector which meets the point M will give us the shortest distance of point M from the plane. 

Graphical Representation of perpendicular distance of a point from a plane

If we express the point of intersection, for example, I, of the line touching the point M, and the plane on which it touches normally, the point I is known as the point which is closest to the point M on the plane. The distance between the two points M and I will give us the distance of the point M from the plane. We can calculate the shortest distance of a point from a plane using two different methods known as:

• The Vector method 
• The Cartesian Method

The video below explains this:

Coordinate Geometry Detailed Video Explanation:

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Vector Form

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For example, let us consider a point B to whom the position vector is given by \(\vec{r}\) and a plane P, which is given by the following equation,

\(\vec{r}. \vec{N} = d\)

where; N is normal to the plane.

Now, consider another plane which is parallel to the first plane and that passes through point B, by this, we will get the equation of the second plane where N is normal to the plane. This equation can be written as

\(\vec{r}-\vec{a}. \vec{N} = 0\) 

or in other easier way,

\(\vec{r} . \vec{N} = \vec{a}.\vec{N}\)

Now let’s consider O as the origin of the coordinates, the distance between the first plane from the origin is written as ON. Similarly, the distance of the other plane from the origin O is denoted as ON’. The distance between these two planes is calculated by ON–ON’. This distance is equal to;

ON−ON′= d′ =|d−a. N^ |

We can also find the perpendicular distance of point B on the plane P’ from P. Therefore, we can conclude that for a plane which is denoted by the following equation;

r^\(\to\). N^\(\to\) = D^\(\to\)

and a point B whose position vector we know, we can calculate the perpendicular distance from a point to the plane with the given formula;

\(d = | \vec{r}.\vec{N} - D| / |\vec{N}|\)

Now if we want to calculate the length of the plane from the origin O, we need to substitute the value 0 in the place of the position vector. This will give us;

d = |D|/ |N|

This is how we calculate the perpendicular distance of a point from a plane in vector form.

Vector Form

Read Also: Different Forms of the Equation of line

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Cartesian Form

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Suppose a plane that is denoted by the Cartesian equation,

Px + Qy + Rz = S

Now suppose that a point whose position vector and Cartesian coordinates of position vector are given by,

A (x₁, y₁, z₁)

We can also write the position vector in the form;

a^\(\to\) = x₁ iˆ+ y₁ jˆ + z₁ kˆ

For finding the distance of point B from the plane by using the vector form of the formula, we can find the normal vector to the plane, which is given as,

N =Aiˆ+ Bjˆ+ Ckˆ

When we use the formula, the perpendicular distance of point B from the given plane is written as,

\(d = | \vec{a}.\vec{N} - D| / |\vec{N}|\)

d = |Ax₁+By₁+Cz₁−D / √A²+B²+C²

This is the equation that will give us the perpendicular distance of a given point from its plane by using the Cartesian method.

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Things to Remember

• The shortest distance covered from one point to a plane is known to be along the line which is perpendicular to the plane, or in other words, is known as the perpendicular distance of that one point from the plane. 
• It is calculated by two different forms namely the vector form, and the cartesian form.
• Formula for calculating the perpendicular distance of a point from a plane in vector form is d = |D| / |N|
• Formula for calculating the perpendicular distance of a point from a plane in cartesian form is d = |Ax₁+By₁+Cz₁−D / √A²+B²+C²|.

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