Angle Measurement Between Lines & Planes

Instructor: Matthew Bergstresser
There is an equation to determine the angle between a line and a plane. In this lesson, we will go through the derivation of the equation and an example using the equation.

Line Intersecting a Plane

If you have ever seen a sundial, you have seen a classic example of a line intersecting a plane. Actually, the part of the sundial that intersects the plane is a plane itself called the gnomon. It is very common for the gnomon to be a right triangle.

Let's pretend the hypotenuse of the right triangle is the line that intersects the circular plane, which is the base of the sundial.

Sundial with the hypotenuse of the gnomon shown in red. This acts as the line that intersects the plane

Let's go through the derivation for the equation to determine the angle between a line intersecting a plane.

Equation Derivation

A mathematical derivation of a geometric scenario usually starts with a diagram so let's make one!

A line intersecting a plane

Here the plane is shaded in gray and the dashed, black line lies on the plane. The dashed, purple arrow is normal to the plane, and the solid, red arrow is the line that intersects the plane. θ is the angle we are looking to find. Φ is the angle between the line and the normal.

There are a couple of steps to determine θ, the angle between the red arrow and the plane. First, we need to determine the angle between the red arrow and the dashed, black line, which we labeled Φ.

We can say there are vectors that lie along the red line and the normal line. A vector has a length and a direction. Since we are switching from two lines to two vectors, we can use an equation to determine Φ, which is:


Let's see how to determine the value in the numerator.


The red vector in our diagram in the notation:


where l, m and n represent the x, y and z components of the vector.

The equation for a plane is:


We don't need the D term in the equation of the plane. All we need from that equation are the A, B and C, which are the coefficients of x, y and z. We'll use the values from the vector and the plane we just discussed to determine the dot product between the vectors. The dot product multiples all x-terms together, y-terms together, and z-terms together and then sums the results. This is what it looks like:


Next, we'll look at how to determine the denominator of our original equation for the angle between two vectors.


The denominator of our equation is the product of the magnitudes of each vector. To determine the magnitude of the vectors, we use the Pythagorean theorem and include the values of all three dimensions. The denominator of our equation is:


Combining the numerator and denominator we get the equation for the angle between the red vector and the vector normal to the plane, which is:


Hmm, we did all of that work and didn't get the angle between the red line and the plane! We got Φ, not θ! But, don't worry, we can get this equation to represent θ, with one quick step. Let's look back at our diagram.


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