It is known that in the cube abcd-a1b1c1d1, e and F are the midpoint of BB1 and CC1 respectively, then the cosine value of the angle between AE and d1f is 0___ ．

It is known that in the cube abcd-a1b1c1d1, e and F are the midpoint of BB1 and CC1 respectively, then the cosine value of the angle between AE and d1f is 0___ ．

Let the edge length of cube abcd-a1b1c1d1 be 2, take DA as x-axis, DC as y-axis, dd1 as z-axis, and establish a space rectangular coordinate system, then a (2, 0, 0), e (2, 2, 1) D1 (0, 0, 2), f (0, 2, 1) ‖ AE = (0, 2, 1), d1f = (0, 2, - 1), let the angle between AE and d1f be θ, then cos θ = | cos < AE, d1f > | = | 0 + 4-15 · = 35

In the cube abcd-a1b1c1d1, e and F are the middle points of edge Aa1 and CC1 respectively, then the line ()
A. There are no B. There are only two C. There are only three d. There are countless D

Take any point m on EF, and the line a1d1 and M determine a plane, which has and only has one intersection n with CD. When m takes different positions, it determines different planes, so it has different intersection n with CD, and the line Mn has intersection with the three different lines. As shown in the figure, select D

It is known that in the cube abcd-a'b'c'd ', e and F are the midpoint of edges BC and CC' respectively. The angle between EF and a'c 'can be calculated

Connect BC 'and a'B; in △ CBC', EF is the median line on BC and c'c, so EF / / BC '①; in △ ba'c', a'B, a'c ', BC' are diagonals of a square, so △ ba'c 'is equilateral, so ∠ bc'a' = 60 °, so according to ①, the included angle of EF and a'c 'is also 60 °

In the cube abcd-a1b1c1d1 with edge length a, M is the midpoint of Aa1, then the distance from point A1 to plane MBD is ()
A. 63aB. 36aC. 34aD. 66a

The distance from a to plane MBD can be obtained by isoproduct deformation. Va-mbd = vb-amd, that is: 112a3 = 13 × D × 12 × 2A × & nbsp; 54a2 − 24a2, that is, it is easy to find d = 66A