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As shown in the figure, in the cube abcd-a1b1c1d1, the tangent of the dihedral angle b-a1c1-b1 is___ ．
Using the vector method, the solution is as follows: take D1 as the origin, d1a1 as the X axis, d1c1 as the Y axis, d1d as the Z axis, establish the d1-xyz space rectangular coordinate system
The volume of cuboid abcd-a1b1c1d1 is 32 and ab = BC = 2a1a = 4, the tangent value of dihedral angle a-d1b1-a1
The root of two is two
Because A1A is perpendicular to plane a1b1c1d1, Aa1 is perpendicular to any line on that plane, Aa1 is perpendicular to b1d1
Take the midpoint m of b1d1 and connect a1m, then:
A1m is perpendicular to b1d1
So triangle ∠ AMA1 is the angle, and the tangent value is Aa1 / a1m = 2 / (2 √ 2) = √ 2 / 2
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