In the cube abcd-a1b1c1d1 with edge length 1, (1) find the distance from point a to plane BD1; (2) find the distance from point A1 to plane ab1d1; (3) find the distance between plane ab1d1 and plane BC1D; (4) find the distance from line AB to cda1b1

(1) Because point a is in plane BD1, the distance between point a and plane BD1 is 0 (2) the volume diagonal of the cube is 3, and the distance between point A1 and plane ab1d1 is 13 of the volume diagonal of the cube; the distance between point A1 and plane ab1d1 is 33; (3) plane ab1d1 ∥ plane BC1D divides the volume diagonal into three equal parts; the distance between plane ab1d1 and plane BC1D is 33; (4) ∵ Ao ⊥ a1d ∥ a to CDA The distance of 1B1 is Ao = 22, while the distance of ab ‖ plane cda1b1 ‖ line AB to cda1b1 is 22

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