Distance from a Point to a Plane

The distance from a point to a plane is the shortest distance, which is the length of the perpendicular line segment from the point to the plane.

1 Formula for Distance

Let the plane be given by its Cartesian equation .
Let the point be . The normal vector to the plane is .

The distance from point to plane is given by:

1.1 Derivation Sketch (Using Scalar Projection)
  1. Let be any point on the plane.
    So, , which means .
  2. Consider the vector .

  3. The distance is the absolute value of the scalar projection of onto the normal vector .

  4. Calculate the dot product:

  5. Substitute :

  6. So,

1.2 Example : Calculating Distance from Point to Plane

Find the distance from the point to the plane .

Solution

Here, . The plane equation gives . The normal vector is .

Distance

The distance is units.

1.3 Example : Point on the Plane

Find the distance from to the plane .

Solution

. .
.
.
Distance .

Since the distance is 0, the point lies on the plane .

1.4 Thinking Question : Distance Between Parallel Planes

How would you find the distance between two parallel planes: (Note: They must have the same coefficients, or be reducible to such, for their normals to be identical, ensuring they are parallel).

Solution

  1. Pick any convenient point on one of the planes, say .
    To do this, you can set two variables to simple values (e.g., ) and solve for the third ( , assuming ).
    So, is a point on .

  2. Calculate the distance from this point to the other plane, , using the standard formula:

    Substitute :

This gives a direct formula for the distance between two parallel planes.

1.5 Example : Distance between parallel planes.

Find the distance between and .

Solution using the derived formula


The distance is 2 units.

Solution by picking a point

Let's find a point on .
Set . Then .
So, is a point on .

Now find the distance from to .

The distance is 2 units. The results match.

2 Homework
2.1 Question 6.1 (Easy)

Find the distance from the point to the plane .

Solution

The formula for the distance from a point to a plane is:

Here, .
The plane is , so .
The normal vector is .

Distance .

Answer: The distance is units.

2.2 Question 6.2 (Medium)

Find the distance between the parallel planes and .

Solution

First, confirm the planes are parallel.

, so they are parallel.
To use the formula where is .

  • Rewrite to have the same normal as :
    Divide by 2:

    So now we compare

The common normal (for this form) is .

Distance (using from and from normalized )

Rationalize: .

Alternative Method: Pick a point on and find its distance to .

Let in :

So, is a point on .
Now find distance from to :

  • .

The results match.

Answer: The distance between the parallel planes is units.

2.3 Question 6.3 (Harder)

Find the coordinates of the point on the plane that is closest to the external point . What is this shortest distance?

Solution

The point on the plane closest to is the foot of the perpendicular from to the plane. The line segment is thus along the normal to the plane.

  • Step 1: Define the line passing through and perpendicular to the plane .
    The normal vector to the plane is .
    This normal vector is the direction vector for the line .
    The line passes through and has direction .
    Parametric equations for :

  • Step 2: Find the intersection of line with plane . This intersection point is .
    Substitute the parametric equations of into the equation of :

    Combine terms:

  • Step 3: Substitute back into the parametric equations of to find the coordinates of .

    So, the point is .

  • Step 4: Calculate the shortest distance, which is the distance between and .
    This can be found using the distance formula for two points, or by using the formula for distance from a point to a plane.
    Using distance formula for and :

Alternatively, using the point-to-plane distance formula for P(3,1,2) and :

  • .

The distances match.

Answer: The point on the plane closest to is . The shortest distance is units.

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