As shown in the figure, in the cube abcd-a1b1c1d1, m, N and P are the midpoint of C1C, b1c1 and c1d1, respectively. We prove that: (1) AP ⊥ Mn; (2) plane MNP ∥ plane a1bd

As shown in the figure, in the cube abcd-a1b1c1d1, m, N and P are the midpoint of C1C, b1c1 and c1d1, respectively. We prove that: (1) AP ⊥ Mn; (2) plane MNP ∥ plane a1bd

It is proved that: (1) connecting BC1 and B1C, then B1C ⊥ BC1, BC1 is the projection of AP on the plane bb1c1c ⊥ AP ⊥ B1C. Then B1C ⊥ Mn, ⊥ AP ⊥ Mn. (2) connecting b1d1, ∵ P and N are the midpoint of d1c1 and b1c1 respectively, ∥ PN ∥ b1d1. Then b1d1 ∥ BD, ∥ PN ∥ BD

In the cube abcd-a1b1c1d1, e and F are the midpoint of AB and CD respectively. What is the tangent value of A1B1 and plane a1ef?
I want a specific problem-solving process, not a train of thought

∵ EF ⊥ aa1b1b
If AG is perpendicular to a1e and G through a, then Ag ⊥ plane a1ef
So eg is the projection of AB in a1ef
The angle between AB and AEF, that is, the angle between A1B1 and a1ef
In △ a1ae, the tangent can be calculated as 2

In the cube abcd-a'b'c'd ', e is the midpoint of d'c', then the tangent of the angle between the line AE and the plane ABCD is

If you pass through point E, make ef perpendicular to CD, perpendicular to foot F, and connect AF, then the angle EAF is the angle formed by AE and plane ABCD Tan angle EAF = EF / AF. if the edge length of cube is 1, then EF = 1, AF = radical (1 & sup2; + (1 / 2) & sup2;) = (radical 5) / 2, so tan angle EAF = 2 / radical 5 = 2 (radical 5) / 5, tangent of the angle formed by AE and plane ABCD

Given that the edge length of cube abcd-a'b'c'd 'is 1, find the tangent of the angle between b'c' and plane ab'c
Such as the title
How to find the vertical line of that plane
If D is the midpoint of a'c ', then the sine value of the angle between the straight line AD and the plane b'dc is?

Related question 1
Because ab '= AC = b'c' = CC ', the vertical ratio of ab'c is in the plane ABC',
Assuming that the midpoint of b'c is m, the vertical line of the extension line of AM can be made through C ', that is, the vertical line of C' perpendicular to the plane ab'c
The tangent value of the angle can be calculated after the length of several sides is calculated