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In the cube abcd-a1b1c1d1 with edge length of 1, Acb1 in the vertical plane of BD1 is proved
Proof: if BD is connected, then
BD⊥AC
Also ∵ dd1 ⊥ ABCD
∴DD1⊥AC
⊥ AC ⊥ bdd1
∴AC⊥BD1
Connecting BA1 is the same as Ab1 ⊥ BD1
{BD1 ⊥ plane Acb1
In the cube abcd-a1b1c1d1, (1) calculate the angle between b1c1 and plane ab1c, (2) calculate the distance between C1 and plane ab1c, (3) calculate the volume of a-b1c1c
Please write the process, thank you
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Establish the coordinate system, a (000) B (100) C (110) d (010) A1 (001) etc. (1) b1c1 vector (010) in plane ab1c, Ab1 vector coordinate (101) AC coordinate (110), then a normal vector of plane ab1c is x (1, - 1, - 1), the sine of the angle is equal to the cosine of the angle between b1c1 and X, equal to - 1
In the cube abcd-a1b1c1d1, there are several sections parallel to AC and passing through the three vertices of the cube
The inclination angle of L1 is A1, and the inclination angle of L2 is A2. In the following four arguments, ① if sina1 = sina2, L1 and L2 coincide; ② if cosa1 = cosa2, L1 and L2 coincide; ③ if cos A1 > cos A2, L1 slope is greater than L2 slope; ④ if Tan A1 > Tan A2, L1 slope is greater than L2 slope
There must be only one, a1b1c1d1, because three points define one plane, and the other planes intersect AC
RELATED INFORMATIONS
- 1. Make a straight line L through the vertex a of cube abcd-a1b1c1d1 so that the angles formed by L and edges AB, ad and Aa1 are equal. Such a straight line l can be made () A. 1 B. 2 C. 3 d. 4
- 2. In the cube abcd-a1b1c1d1, M is the midpoint of dd1, O is the midpoint of AC. AB = 2. Verification: b1o ⊥ plane ACM. Can you tell me
- 3. In trapezoidal ABCD, ab ∥ DC, diagonal AC, BD intersect with O, through o as Mn ∥ DC intersects ad with m, intersects BC with N. to prove the value of OM / AB + on / BC AB is the bottom, DC is the bottom, a and D are on the left
- 4. Given the edge length of cube abcd-a'b'c'd 'as a, find the angle a'B and b'c, a'B vertical AC'
- 5. In the cube ABCD-A ′ B ′ C ′ D ′ with edge length 1, AC ′ is diagonal, M. N is the midpoint of BB ′, B ′, C ′, P is the midpoint of Mn (1) Finding the tangent of the angle between DP and ABCD (2) Find the tangent of the angle between DP and AC '
- 6. 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___ ．
- 7. As shown in the figure, the edge length of cube abcd-a1b1c1d1 is 8, m, N, P are the midpoint of A1B1, ad, B & nbsp; B1 respectively. (1) draw the intersection line between the plane passing through point m, N, P and plane ABCD and the intersection line with plane bb1c1c1c; (2) let the plane PMN and edge BC intersect at point Q, and calculate the length of PQ
- 8. In the cube ABCD ABCD with edge length 1, m and N are the midpoint of the line segments BB and BC respectively, and the distance between the line Mn and ACD is calculated
- 9. 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
- 10. The cosine value of the dihedral angle b-dc '- C is? I take d 'as the origin to build the space right-handed system, and the result shows that the normal vector of DCC' is (0,0,0), which is unscientific!
- 11. As shown in the figure, e is the midpoint of dd1 in cube abcb-a1b1c1d1. Try to judge the position relationship between BD1 and plane AEC, and explain the reason
- 12. In the pyramid p-abcd as shown in the figure, the bottom surface ABCD is rhombic, the PA ⊥ plane ABCD, and E is the midpoint of PC. It is proved that: (1) Pa ⊥ plane BDE; & nbsp; & nbsp; (2) plane PAC ⊥ plane PBD
- 13. In the cube abcd-a'b'c'd ', P and Q are the midpoint of a'B', BB ', respectively Find the angle between AP and CQ and the angle between AP and BD Let the edge length of cube be 2 (1) Take the midpoint m of AB and the midpoint n of CC 'to connect B'm and B'n Then: the angle mb'n is the angle formed by the line AP and CQ B'M=B'N=√5,MN=√6 From the cosine theorem, it is concluded that cos(MB'N)=2/5 Angle mb'n = arccos (2 / 5) (2) Connect b'd ', then the angle mb'd' is the angle formed by the line AP and BD B'D'=2√2,D'M=3 From the cosine theorem, it is concluded that: Cos (MB'd ') = from cosine theorem: cos(MB'D')=√10/10 Angle MB'd '= arccos (√ 10 / 10) Why is d'm equal to 3?
- 14. In the parallel hexahedron abcd-a1b1c1d1, the In the parallelepiped abcd-a1b1c1d1, the lengths of the three edges with vertex a as the endpoint are all 1, and their included angles are all 60 ° to each other. The distance between the plane ABCD and a1b1c1d1 is calculated
- 15. It is known that the three edges of the parallelepiped abcd-a1b1c1d1 with the same vertex a as the end point are all equal to 1 and the angle between them is 60 degrees, and the length of the ball AC1
- 16. In the cube ABCD-A 'B' C'd ', if the midpoint of BC is e and the midpoint of CD is f, then the angle between ad' and EF is
- 17. As shown in the figure, in the cube abcd-a1b1c1d1, e and F are the midpoint of AB and Aa1 respectively
- 18. In the cube abcd-a'b'c'd 'with edge length a, the plane a'bd / / plane cb'd' is proved
- 19. What is the figure obtained by rotating one circle around the axis of the straight line of an edge of the cube?
- 20. In the tetrahedral ABCD, ab = a, CD = B, the distance between AB and CD is d. when the angle θ of edge AB and CD is what, what is the maximum volume?