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.

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:

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,

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:

Problems related to the determination of angles between two planes:

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

Solution:  As 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\(\hat{k}\)