Angle Between Two Planes - Planes & Angles

In 3d geometry, position vectors are used to denote the position or location of a point with respect to the origin. The plane, as we know, is a 3d object formed by stacks of lines kept side by side. A 3d space can have an infinite number of planes aligned to one another at an infinite number of angles. In this article, we will derive a general formula for the calculation of angle between two planes in 3d space.

Calculation of angle between two planes:

The angle between two planes is generally calculated with the knowledge of angle between their normal. In other words, the angle between normal to two planes is the angle between the two planes. This can be understood quite clearly from the below figure:

angle between two planes

Let \(\vec{n_{1}}\) and \(\vec{n_{2}}\) be the two normal to the planes aligned to each other at an angle θ. The equation of two planes can be given by:
\(\vec{r}\).\(\vec{n_{1}}\) =  d1
\(\vec{r}\).\(\vec{n_{2}}\) =  d2

From the above figure, we learnt that the angle between the two planes is equal to the angle between their normal, thus,

angle between two planes

Calculation of angle between two planes in the Cartesian form:

Let A1 x + B1 y + C1z + D1 = 0 and A2x + B2y + C2 z + D2 = 0  be the equation of two planes aligned to each other at an angle θ where A1, B1, C1 and A2, B2, C2 are the direction ratios of the normal to the planes.

The cosine of the angle between the two planes is given by:

Cartesian form

Problems related to the determination of angles between two planes:

QuestionCalculate the angle between the two planes given by the equation 2x + 4y – 2z = 5 and 6x – 8y – 2z = 14.

SolutionAs learnt above, the angle between two planes is equal to the angle between their normals. Normal vectors to the above planes are represented by:

\(\vec{n_{1}}\) = 2\(\hat{i}\) + 4\(\hat{j}\) – 2\(\hat{k}\)
\(\vec{n_{2}}\) = 6\(\hat{i}\) – 8\(\hat{j}\) – 2\(\hat{k}\)

angle between two planes

angle between two planes

To learn about other techniques to calculate the angle between two planes downloads Byju’s- The Learning App.


Practise This Question

In a hypothetical universe, all the numbers are represented using dots. Each number is represented by that many dots as the value of the number. These dots are then joined by a straight line. The number 2 when represented in this world, would only form a ___.