When calculating dihedral angle with vector method, first calculate the included angle of vector, and then deduce the size of dihedral angle.

However, the normal vector of a plane may have two directions by definition (as shown in figure 1 and n 1' of the α plane).

In order to find the normal vector angle, there can be:

(1) is the included angle between n 1 and n2 as shown in Figure 2 (= the included angle between n1'and n2'), which can be easily proved to be a dihedral complementary angle.

(2) As shown in Figure 3, it is n 1 and N2' (the angle between n = n1'and N2), which is a dihedral angle.

So sometimes it is a dihedral angle, and sometimes it is its complementary angle.

(2)

If the included angle of shilling normal vector is θ, cosθ can be obtained from the formula.

Then through observation, reasoning or understanding, it is concluded that dihedral angle is acute or obtuse.

(1) If the dihedral angle is acute, its cosine value should be positive.

If cosθ is positive, θ is a dihedral angle.

If cosθ is negative, θ is the complementary angle of dihedral angle.

Taking a graph as an example, the dihedral angle is an acute angle through observation. If the vectors n 1 and n2 are calculated, then cosθ is negative and dihedral angle is π-θ.

If the vectors n 1 and N2' are calculated, then cosθ is positive and dihedral angle is θ.

(2) If the dihedral angle is obtuse, the other chord values should be negative. Similarly:

If cosθ is positive, θ is the complementary angle of dihedral angle.

If cosθ is negative, θ is a dihedral angle.

(3)

Use vector to calculate dihedral angle, first calculate vector,

You can control the direction of the normal vector as shown in Figure 3 (that is, one point is inside the dihedral angle and the other point is outside the dihedral angle), so you can directly take θ as the dihedral angle.

You can also make the normal vector direction arbitrary, as long as you judge whether θ is a dihedral angle or its complementary angle as in (2).

To find the normal vector, you can use the undetermined coefficient method (refer to Baidu Encyclopedia Normal Vector).

You can also cross and multiply the vectors of two different straight lines in the plane (if you don't master it in high school, you'd better use the former).