In this explainer, we will learn how to calculate the perpendicular distance between a plane and a point, between a plane and a straight line parallel to it, and between two parallel planes using a formula.
To find the shortest distance between a point and a line, we first need to determine exactly what is meant by the shortest distance between these two geometric objects. To do this, we first note that if a point lies on the plane , then the distance between these objects will be zero. So, we will assume our point does not lie on the plane.
To find the shortest distance between these objects, let us first consider the distance between and a point on our plane.
We can show that this is not the shortest distance between and the plane by constructing the following right triangle.
We choose point on our plane so that the line segment is perpendicular to the plane. We can then see that is the hypotenuse of a right triangle, which means it must be longer than the other sides. In particular, this means that the length of is shorter than . We can construct this triangle for any point on our plane, so the must be the shortest distance between the point and the plane.
We call this the perpendicular distance between the point and the plane because is perpendicular to the plane. We could find this distance by finding the coordinates of ; however, there is an easier method.
To calculate this distance, we will start by setting and . We will also introduce the vectors and .
We see in our diagram that is the length of the side adjacent to angle in a right triangle; this tells us that
In particular, we get the equation
We can construct another equation involving the expression by recalling the following property about vectors.
Definition: The Dot Product of Two Vectors
If is the measure of the angle between two vectors and , then
Applying this property to the vectors and , we get
We can rearrange this equation to get
However, we cannot evaluate this expression directly because we do not know the coordinates of . We can get around this by recalling that is perpendicular to the plane and we can find another vector perpendicular to the plane.
Recall that the vector is perpendicular to the plane . Therefore, since both and are perpendicular to the plane, we must have .
Remember, we say two vectors are parallel if they are a non zero scalar multiples of each other; we will call this scalar :
We can use this to find the perpendicular distance . First, we can substitute this expression into our equation for the dot product and then simplify:
Next, we need to be careful to simplify , since we do not know if is negative or positive. Instead, we know that is a length and must therefore be positive. This means we can take the absolute value of both sides of this equation:
We can then simplify this equation by using the properties of the absolute value:
We could leave our expression for in this form; however, we can simplify this further. Recall that is any point on our plane; let us say .
We can then find the components of :
Then, we can substitute this expression into our equation for and evaluate the dot product:
Finally, we will use the fact that lies on the plane ; this means
We can rearrange this to see
Substituting this into our equation for and simplifying, we get
We can summarize this result as follows.
Definition: Distance between a Point and a Plane
The shortest distance (or perpendicular distance), , between the point and the plane is given by
Let us see an example of how we can use this formula to find the perpendicular distance between a point and a plane given in the general form.
Example 1: Finding the Distance between a Point and a Plane
Find the distance between the point and the plane .
Answer
We want to find the distance between a point and a plane. To do this, we first need to recall that the distance between a point and a plane means the perpendicular distance, since this is the shortest distance between these two objects.
To find the perpendicular distance, we need to recall the following formula.
The perpendicular distance, , between the point and the plane is given by
We have and we need to rewrite our equation for the plane
So, , , , and .
Substituting these values into our formula, we get
We can add length units to this value since it represents a length.
Hence, we were able to show that the distance between the point and the plane is .
In our next example, we will see how we can apply this formula to find the distance between a point and a plane given in vector form.
Example 2: Finding the Distance between a Point and a Plane
Find the distance between the point and the plane .
Answer
We want to find the distance between a point and a plane. To do this, we need to recall that the distance, , between the point and the plane is given by
We cannot apply this formula directly because our plane is given in vector form. Therefore, to apply our formula we want to convert the plane into the general form for the equation of our plane.
To do this, we substitute into the vector equation of our plane:
Then, we subtract 3 from both sides of the equation:
Now that we have the equation of our plane in the general form, we can apply our formula for the distance. We have , , , , and ; substituting these values in, we get we can add length units to this value because we know it represents a length.
Hence, we were able to show that the distance between the point and the plane is .
In the previous example, we found the distance between a point and a plane given in the vector form by finding the equation of our plane in the general form. We can use this process to find a formula for the distance between a point and a plane in the vector form.
We can always rewrite the plane with equation into the form . Applying our formula for the perpendicular distance gives us the following result.
Theorem: Distance between a Point and a Plane in Vector Form
The shortest distance (or perpendicular distance), , between the point and the plane is given by
We could have used this result to directly evaluate the distance given to us in the second example.
We can use this same process to determine the distance between a line and a plane. First, if the line and plane are not parallel or not distinct, then they intersect, so the distance between them is 0. Second, if they are parallel and distinct then, we can show that the shortest distance between them is the perpendicular distance between any point on the line and the plane. Consider the distance between an arbitrary point on a line and another arbitrary point on the plane parallel to the line.
We can show that is the hypotenuse of a right triangle, so this distance is always larger than the perpendicular distance between the point and the plane. Finally, since the line and plane are parallel, the distance between them is constant, so we can choose any point on our line and the distance will be the same, which means we can use the formula for the distance between a point and a plane.
Theorem: Distance between a Line and a Plane in Vector Form
The shortest distance (or perpendicular distance), , between a parallel line and a plane, where is any point on the line and the plane has the equation , is given by
In our next two examples, we will see how we can apply this process to find the distance between a line parallel to a plane and said plane.
Example 3: Finding the Distance between a Line and a Plane
Find the perpendicular distance between the line and the plane .
Answer
The question asks us to find the perpendicular distance between a line and a plane. We need to determine if they intersect; to do this, we first rewrite the line as
Then, we substitute this into the equation of the plane:
This equation will not be true for any value of , so the line and plane do not intersect. Hence, they are parallel.
Alternatively, we can show that the line and plane are parallel by showing that the normal vector to the plane and the direction vector of the line are perpendicular; we can do this by computing their dot product:
Hence, the line is perpendicular to the normal vector of the plane, and so the line and plane are parallel.
We recall that the distance between a line and a plane is given by the distance between any point on the line and the plane. We know that the point lies on the line, since this is the position vector when , and that the perpendicular distance, , between the point and the plane is given by
We substitute , , , , , , and into the formula to get
Hence, the distance between the line and plane, to one decimal place, is length units.
Example 4: Finding the Distance between a Line and a Plane
Find the distance between the line and the plane . Give your answer to one decimal place.
Answer
The question asks us to find the perpendicular distance between a line and a plane. We need to determine whether they intersect. We first check if the line and plane are parallel. For the line and plane to be parallel, the direction vector of the line must be perpendicular to the normal vector of the plane. We can check this by computing their dot product. The direction vector of the line is and the normal vector of the plane is , giving us
Since this is equal to zero, the line is perpendicular to the normal vector of the plane, which means they are parallel.
We recall that the distance between a parallel line and a plane is the same as the distance between any point on the line and the plane. Setting each part of the Cartesian equation of the line equal to zero and solving tells us that lies on the line. We also know that the distance, , between the point and the plane is given by
Substituting , , , , , , and into this formula gives us
Hence, the distance between the line and plane to one decimal place is 1.1 length units.
We can also use the formulae to determine the distance between two parallel planes. To do this, we could try finding the distance between an arbitrary point on each plane, which we will call and .
However, if we compare this to the perpendicular distance, we can see that is the hypotenuse of a right triangle, which means it is longer than the perpendicular distance. This is true for any two points we choose. In other words, the shortest distance between two parallel planes is the perpendicular distance. In fact, since parallel planes stay the same distance apart, we can choose any point to be our starting point. Thus, we can use our formulae for the distance between a point and a plane to determine the distance between two parallel planes.
In our final example, we will see how to apply this process to find the distance between two parallel planes.
Example 5: Finding the Distance between Two Planes
Find the distance between the two planes and .
Answer
We want to find the distance between two planes. To do this, we will start by checking whether the two planes are parallel, since we can then apply the formula for the perpendicular distance.
We recall that two planes are parallel if the normal vectors to each plane are parallel. The normal vector to each plane is given by the coefficients, so the normal vectors of the two planes are and , which are scalar multiples of each other. Hence, the planes are parallel.
We want to find a point on one of our planes; to do this, we can substitute and into the equation of our first plane:
This means that the point lies on the first plane. To find the distance between the two planes, we will find the distance between the point and the plane .
We recall that the perpendicular distance, , between the point and the plane is given by
To apply this, we need to rewrite the equation of our plane by subtracting 3 from both sides of the equation:
This gives us , , , and . Substituting these values and our point into our formula gives us
We can add length units to this value because we know it represents a length.
Hence, we were able to show that the distance between two planes and is .
Let us finish by recapping some of the important points of this explainer.
Key Points
- The distance, , between the point and the plane is given by
- The distance, , between the point and the plane is given by
- The distance between a line parallel to a plane and said plane is equal to the distance between any point on the line and the plane.
- The distance between two parallel planes is equal to the distance between any point on either plane and the other plane.
- The perpendicular distance between a point and a plane is the shortest distance between these two objects.
