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Maths > Three Dimensional Geometry > Angle Between Two Planes
Three Dimensional Geometry

Angle Between Two Planes

The angle between two planes is the angle between the normal to the two planes. Read this lesson on Three Dimensional Geometry to understand how the angle between two planes is calculated in Vector form and in Cartesian form. The understanding of the angle between the normal to two planes is made simple with a diagram. A solved example, in the end, is also explained to understand how the calculation is performed.

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3-D Geometry: The Plane

In Mathematics, ‘planes’ form an important part of 3-D geometry. What is a plane? It is a two-dimensional figure extending infinitely in the three-dimensional space but has no thickness. You can imagine a plane to be an extended number of lines arranged together side by side in the three-dimensional space.

Note that an infinite number of planes can exist in the three-dimensional space. In coordinate geometry, we use position vectors to indicate where a point lies with respect to the origin (0,0,0). Let us now move to how the angle between two planes is calculated.

Browse more Topics Under Three Dimensional Geometry

The Angle Between Two Planes

Just like the angle between a straight line and a plane, when we say that the angle between two planes is to be calculated, we actually mean the angle between their respective normals. Thus, we are now actually going to learn how the angle between the normal to two planes is calculated. A close look at the figure below explains this clearly.

angle between two planes

Source: The Learning Point

Calculation in Vector Form

Let us consider two planes, let the normal to the planes be n1 and n2 respectively. The equations of the planes can be written as:

  • r.n1 = d1
  • r.n2 = d2

Then, the cosine of the angle (between the normal to both planes is given by:

Cos  = | n1.n2 | / |n1|. |n­2 |, where the modulus in the denominator refers to the magnitude of the vectors.

Calculation in Cartesian Form

In the Cartesian form, the equation of two planes may be written as a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0. Let us consider as the angle between the normal to the two planes and (a1, b1, c1) & (a2, b2, c2) are the direction ratios of the normal to both the planes in consideration.

Again, the cosine of the angle between the two planes can be given by:

Cos  = | a1a2 + b1b2 + c1c2 | / (a12 + b12 + c12)1/2 (a22 + b22 + c22)1/2

The following example shall help you understand the calculation better.

Solved Example for You

Question 1: Find the angle between the planes whose vector equations are given by r. (2i + 2j – 3k) = 5 and r. (3i – 3j + 5k) = 3.

Answer: We can see that the problem is given in vector form, so we will use the formula in vector form to calculate the angle between the two planes. Comparing with the general equation of a plane in vector form,

n1 = 2i + 2j – 3k and n2 = 3i – 3j + 5k, while

| n1 | = (22 + 22 + (-3)2)1/2 = 171/2 and | n2| = (32 + (-3)2 + 52)1/2 = 431/2.

Thus, Cos = (2i + 2j – 3k). (3i – 3j + 5k) / 171/2. 431/2

Cos = | 2×3 + 2x(-3) + (-3)x5 | / 171/2. 431/2

Cos = | 6 -6 – 15 | / 171/2. 431/2

Cos = | -15 | / 7311/2

Cos = 15 / 7311/2

So,  = Cos-1 (15 / 7311/2)

Question 2: What is the measure of a dihedral angle?

Answer: A dihedral angle refers to the angle between two planes. Remember, that a plane refers to a flat two-dimensional surface. We notice that each of the sides of the cube will be a plane and the angles in-between each of these planes will be 90 degrees. Therefore the dihedral angles of a cube will each be 90 degrees

Question 3: What does the dihedral angle mean?

Answer: A dihedral angle refers to the angle that is between two intersecting planes. In chemistry, it refers to the angle which is between planes through two sets of three atoms, which has two atoms in common. In solid geometry, we define it as the union of a line and two half-planes that are having this line as a common edge.

Question 4: Why is the dihedral angle important?

Answer: The function of dihedral effect is to give stability in the roll axis. Moreover, it is a significant factor in the constancy of the spiral mode that is occasionally referred to as “roll stability”.

Question 5: What is the dot product of two vectors?

Answer: In Algebra, the dot product refers to the sum of the products of the consequent entries of the two sequences of numbers. In geometry, it refers to the product of the Euclidean magnitudes of the two vectors as well as the cosine of the angle that is between them. Thus, these definitions are correspondent when we use Cartesian coordinates.

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