Fold △BCD into △BC'D along the straight line BD.
It can be seen that CD=6, BC'=BC= 10, BD=8,
That is BC'2 = C'D2+BD2,
Therefore, C.D. ⊥ BD ... (2 points)
∫ plane BC'D⊥ plane ABD, plane BC'D∩ plane ABD=BD, C'D? Planes in BC,
∴C'D⊥ Aircraft Abd ... (5 points)
(ii) According to (i), the plane ABD with known C'D⊥ and the CD⊥BD,
As shown in the figure, the spatial rectangular coordinate system D-XYZ is established with D as the origin. ... (6 points)
Then d (0 0,0,0), a (8 8,6,0), b (8 8,0,0), c' (0,0,6).
E is the midpoint of the AD line,
∴E(4,3,0),BD=(? 8,0,0).
On BEC's plane, be = (? 4,3,0),BC′=(? 8,0,6),
Let the normal vector of plane BEC be n = (x, y, z),
∴BE? n = 0BC′? N = 0, that is? 4x+3y=0? 8x+6z=0,
Let x=3, y=4 and z=4, so n = (3 3,4,4) ... (8 points).
Let the included angle between straight line BD and plane BEC be θ, then sin θ = | cos < n, BD > | = | n? BD||n|? |BD|=34 14 1。 ? ... (9 points)
∴ The sine value of the angle formed by the straight line BD and the plane BEC' is 34 14 1.