In this explainer, we will learn how to find the measure of an angle between two planes or that between a line and a plane.

Let us consider two intersecting planes and .

The angle between them can be visualized in a plane perpendicular to their line of intersection. We see that, as for two intersecting lines, there are two angles between them, and , with .

Drawing now the two pairs of possible normal vectors of the two planes, we see that either or is the angle between the normal vectors, depending on the directions of the normal vectors.

If we define the angle between two planes and as the acute angle between them, that is, , then we can write since the scalar product with and is and (because ).

### Definition: Angle between Two Planes

The angle between two planes and with normal vectors and is defined as the acute angle between them; hence, , and we have

Let us look at the first example.

### Example 1: Finding the Angle between Two Planes given Their General Equations

Find, to the nearest second, the measure of the angle between the planes and .

### Answer

To find the angle between the planes, we need the normal vectors of the two planes. Knowing that the general equations of a plane are of the form , with being the components of a normal vector of the plane, we find that the components of the normal vectors of the two planes are and . The angle between the two planes is such that where and are two normal vectors of the two planes.

We have

To find , we use the inverse function on our calculator:

We need to give the angle to the nearest second. Remember that one degree is 60 minutes and one minute is 60 seconds .

Therefore, and

The angle between the planes is .

Let us now find the angle between two planes when the equation of one of the planes is a general equation and the other is a vector equation.

### Example 2: Finding the Angle between Two Planes given Their Standard and Vector Equations

Find, to the nearest degree, the measure of the angle between the planes and .

### Answer

To find the angle between the planes, we need the normal vectors of the two planes. The coefficients of , , and in the general equation are the -, -, and -components of a normal vector of the plane. In the equation given, we see that expanding the brackets will each time give a term in , , or and a constant. The coefficients of , , and are therefore given by the coefficient in front of each bracket. So, a normal vector of the first plane is .

The other plane equation is given in vector form, where is a normal vector of the plane.

The angle between the two planes is such that where and are two normal vectors of the two planes. Hence, and

The angle between the two planes to the nearest degree is .

Consider now a line intersecting a plane .

The angle between line and plane is defined as the smallest possible angle between line and any line in plane that intersects (i.e., that goes through the intersection point of with ). Hence, it is the acute angle between line and intersection line between plane and plane , which is the plane perpendicular to that contains . It is therefore defined as the complementary angle to the smallest possible angle between line and the normal of plane , as we can visualize in plane shown in the next figure.

The direction vector of and the normal vector of are also represented. The vector indicated with a dashed line is a normal vector of with an opposite direction to that of .

We find that the angle between line and plane is either or , where and are the two possible angles between and , depending on their respective directions. (Note that we would find angle between as shown here and the direction vector of in the opposite direction to as shown in the diagram.) However, as for the angle between two planes, we know that will give an acute angle (here ), since and have the same absolute value but is negative while is positive.

Therefore, angle between line and plane is complementary to , with .

As , it follows that

### Definition: Angle between a Line and a Plane

Angle between a plane with normal vector and a line with direction vector is defined as the complementary angle to the smallest possible angle between line and the normal of the plane .

It is such that

We can easily see in a right triangle why if , then .

However, note that and are true equalities beyond the case of right triangles (i.e., also for angles ) because of the symmetries of the sine and cosine functions, as illustrated here with unit circles. Point associated with angle is the reflection in line of point associated with angle , meaning that and ; therefore, and .

Let us see in the next example how to find the angle between a plane and a line given their vector equations.

### Example 3: Finding the Angle between a Plane and a Line given their Vector Equations

Which of the following is the smaller angle between the straight line and the plane ?

### Answer

To find the angle between a line and a plane, we need to know the components of a direction vector of the line and of a normal vector of the plane. In a vector equation of a line, the direction vector is the vector that is multiplied by . Here, it is ; its components are therefore .

The vector equation of a plane is of the form , where is a constant. Here, the normal vector of the plane is ; its components are .

The angle between a plane with normal vector and a line with direction vector is such that

Let us calculate :

The value is the value of ; therefore, . The correct answer is C.

Let us apply the same idea for the last two examples but with different types of line equations.

### Example 4: Finding the Angle between a Plane and a Line given their General and Parametric Equations

Find, to the nearest second, the measure of the angle between the straight line , , and the plane .

### Answer

To find the angle between a line and a plane, we need to know the components of a direction vector of the line and of a normal vector of the plane. In the parametric equations of a line, the -, -, and -components of the direction vector are the coefficient of in the equation for , , and respectively. We find that the components of the direction vector of the given line are .

The general equation of a plane is of the form , with being the components of a normal vector of the plane. Hence, the components of the normal vector of the plane here are .

The angle between a plane with normal vector and a line with direction vector is such that

Let us calculate :

To find the value of , we use on our calculator the inverse function of sine:

We need to give the angle to the nearest second. Remember that one degree is 60 minutes and one minute is 60 seconds .

Therefore, and

The angle between the planes is .

### Example 5: Finding the Angle between a Plane and a Line given their General and Cartesian Equations

Find, to the nearest second, the measure of the smaller angle between the straight line and the plane .

### Answer

To find the angle between a line and a plane, we need to know the components of a direction vector of the line and of a normal vector of the plane. The Cartesian equation of a line is of the form where are the coordinates of a point that lies on the line and is a direction vector of the line. Hence, the components of the direction vector of the given line are .

The general equation of a plane is of the form , with being the components of a normal vector of the plane. Hence, the components of the normal vector of the plane here are .

Angle between a plane with normal vector and a line with direction vector is such that

Let us calculate :

To find the value of , we use on our calculator the inverse function of sine:

We need to give the angle to the nearest second. Remember that one degree is 60 minutes and one minute is 60 second .

Therefore, and

The angle between the planes is .

### Key Points

- Angle between two planes and with normal vectors and is defined as the acute angle between them; hence, , and we have
- Angle between a plane with normal vector and a line with direction vector is such that