How is the dihedral angle found?

Let the normal vectors of the two planes be n1,n2.Let the plane angle of the dihedral angle be α.

Then cosα=±? (n1•n2)? /? (|n1||n2|)

Taking a positive or negative sign depends on whether this dihedral angle is acute or obtuse.

How do you determine whether it is acute or obtuse? There are two ways:

Based on the topic, it is obvious whether it is acute or obtuse, at which point the sign is taken positive or negative;

Based on the direction of the two normal vectors: the dihedral angle divides space into two parts. When the two normal vectors cross the plane, if the direction points to the same side, then take a negative sign, if the direction points to the opposite side, ? then the positive sign is taken.

First set up the appropriate right-angle coordinate system in space, label the coordinates of each point, let the dihedral angle be A-BC-D, let the two normal vectors be m(a,b,c),n(x,y,z), then we have

m*AB=0

m*BC=0

n*BD=0

n*CD=0

Take the appropriate m and n values,find m*n/|m|*|n|. Depending on the direction of m and n you take, if they both point inward or outward of the dihedral angle, the dihedral size is

1-arccos(m*n/|m|*|n|); if one points inward and one points outward, the dihedral size is arccos(m*n/|m|*|n|).

It is known that the normal vectors of the two faces of the dihedral angle α-l-β are m vector=(-1,0,2) n vector=(3,-1,0) The sine of this dihedral angle is ?

Let the angle between the two normal vectors be a,

cosa=mn/|m||n|

=-1*3+0*(-1)+2*0/root [((-1)^2+0+2^2)(3^2+(-1)^2+0]

=-3/(5root2)

Because there is a quadrilateral with angle sum of π ,and the remaining two are right angles,

it can be seen that the dihedral angle is complementary to the angle of a. Let the dihedral angle be b

Then sinb=sin(π-a)=sina=root[1-(cosa)^2]=root(41/50)=root(82)/10