Angle Between Two Planes

Let the planes be given by the general equations

The angle between two planes is defined as the angle between their respective normal vectors.

Once you have the normal vectors for the two planes, you can compute the dot product of these vectors. The dot product of two vectors n₁ and n₂ is given by

where φ is the angle between the vectors and, accordingly, the dihedral angle between the planes.

Using the dot product formula, you can solve for φ and express the cosine of φ in coordinate form:

Finally, you can find the angle φ by taking the arccosine (inverse cosine) of the result.

Solved Problems

Solution.

We use the formula

Let's substitute the known coefficients:

Then the angle φ between the planes is equal to

Solution.

If a plane passes through the $$axis then its equation has the form

Let's write down what is the angle $$ between the two planes:

Since $$ we get the following relation:

We just need to find the ratio of coefficients $$ and $$ So we put $$ and get

Let's square both sides of the equation and find $$

Find the roots of the quadratic equation:

So we got two sets of coefficients $$ and $$ that correspond to two possible planes:

Solution.

Let point $$ be an arbitrary point of our plane. The distance from point $$ to two given planes must be the same. Therefore we can write the following equation

Substituting the coefficients we get

or

This equation contains two solutions. In the first case we have

The second solution, that is, another plane is described by the equation

Solution.

The base $$ lies in the $$plane, so its equation is $$

Let's find the equation of the $$ face given three points:

Substitute the coordinates of points $$ $$ $$

Expand the determinant along the first row:

Find the angle between planes $$ and $$ using the formula

Hence

Since cosine is negative, we have found an obtuse angle $$ between the planes. The corresponding acute angle $$ between the planes is equal to