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幾何沒方向,快去請向量(2019課標(biāo)Ⅰ,12)

2023-01-25 19:35 作者:數(shù)學(xué)老頑童  | 我要投稿

(2019課標(biāo)Ⅰ,12)已知三棱錐P-ABC的四個頂點(diǎn)在球O的表面上,PA%3DPB%3DPC,%5Cbigtriangleup%20ABC是邊長為2的等邊三角形,E、F分別是PA、AB的中點(diǎn),%5Cangle%20CEF%3D90%5E%5Ccirc%20,則球O的體積為(? ? )

A.8%5Csqrt%7B6%7D%5Cmathrm%7B%5Cpi%7D

B.4%5Csqrt%7B6%7D%5Cmathrm%7B%5Cpi%7D

C.2%5Csqrt%7B6%7D%5Cmathrm%7B%5Cpi%7D

D.%5Csqrt%7B6%7D%5Cmathrm%7B%5Cpi%7D

解:

設(shè)%5Cleft%5C%7B%20%5Coverrightarrow%7BPA%7D%2C%5Coverrightarrow%7BPB%7D%2C%5Coverrightarrow%7BPC%7D%20%5Cright%5C%7D%20為空間中所有向量的一組基底,其中

%5Cleft%7C%20%5Coverrightarrow%7BPA%7D%20%5Cright%7C%3D%5Cleft%7C%20%5Coverrightarrow%7BPB%7D%20%5Cright%7C%3D%5Cleft%7C%20%5Coverrightarrow%7BPC%7D%20%5Cright%7C%3Dn,

%5Cleft%3C%20%5Coverrightarrow%7BPA%7D%2C%5Coverrightarrow%7BPB%7D%20%5Cright%3E%20%3D%5Cleft%3C%20%5Coverrightarrow%7BPB%7D%2C%5Coverrightarrow%7BPC%7D%20%5Cright%3E%20%3D%5Cleft%3C%20%5Coverrightarrow%7BPC%7D%2C%5Coverrightarrow%7BPA%7D%20%5Cright%3E%20%3D%5Ctheta%20

%5Coverrightarrow%7BCE%7D%3D%5Coverrightarrow%7BPE%7D-%5Coverrightarrow%7BPC%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7BPA%7D-%5Coverrightarrow%7BPC%7D

%5Coverrightarrow%7BEF%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7BPB%7D,則

%5Cbegin%7Baligned%7D%0A%09%5Coverrightarrow%7BCE%7D%5Ccdot%20%5Coverrightarrow%7BEF%7D%26%3D%5Cleft(%20%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7BPA%7D-%5Coverrightarrow%7BPC%7D%20%5Cright)%20%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7BPB%7D%5C%5C%0A%09%26%3D%5Cfrac%7B1%7D%7B4%7D%5Coverrightarrow%7BPA%7D%5Ccdot%20%5Coverrightarrow%7BPB%7D-%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7BPB%7D%5Ccdot%20%5Coverrightarrow%7BPC%7D%5C%5C%0A%09%26%3D%5Cfrac%7B1%7D%7B4%7D%5Cleft%7C%20%5Coverrightarrow%7BPA%7D%20%5Cright%7C%5Ccdot%20%5Cleft%7C%20%5Coverrightarrow%7BPB%7D%20%5Cright%7C%5Ccdot%20%5Ccos%20%5Ctheta%20-%5Cfrac%7B1%7D%7B2%7D%5Cleft%7C%20%5Coverrightarrow%7BPB%7D%20%5Cright%7C%5Ccdot%20%5Cleft%7C%20%5Coverrightarrow%7BPC%7D%20%5Cright%7C%5Ccdot%20%5Ccos%20%5Ctheta%5C%5C%0A%09%26%3D%5Cfrac%7B1%7D%7B4%7Dn%5Ccdot%20n%5Ccdot%20%5Ccos%20%5Ctheta%20-%5Cfrac%7B1%7D%7B2%7Dn%5Ccdot%20n%5Ccdot%20%5Ccos%20%5Ctheta%5C%5C%0A%09%26%3D-%5Cfrac%7B1%7D%7B4%7Dn%5E2%5Ccos%20%5Ctheta%20%3D0%5C%5C%0A%5Cend%7Baligned%7D

所以%5Ccos%20%5Ctheta%20%3D0,

所以%5Ctheta%20%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B%5Cmathrm%7B%5Cpi%7D%7D%7B2%7D%7D

%5Cleft%7C%20PA%20%5Cright%7C%3D%5Cleft%7C%20PB%20%5Cright%7C%3D%5Cleft%7C%20PC%20%5Cright%7C%3D%5Ccolor%7Bred%7D%7B%5Csqrt%7B2%7D%7D

所外接球的半徑

R%3D%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20%5Csqrt%7B%5Cleft(%20%5Csqrt%7B2%7D%20%5Cright)%20%5E2%2B%5Cleft(%20%5Csqrt%7B2%7D%20%5Cright)%20%5E2%2B%5Cleft(%20%5Csqrt%7B2%7D%20%5Cright)%20%5E2%7D%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B%5Csqrt%7B6%7D%7D%7B2%7D%7D

所以外接球的體積

V_%7B%5Ctext%7B%E7%90%83%7D%7D%3D%5Cfrac%7B4%7D%7B3%7D%5Cmathrm%7B%5Cpi%7D%5Ctimes%20%5Cleft(%20%5Cfrac%7B%5Csqrt%7B6%7D%7D%7B2%7D%20%5Cright)%20%5E3%3D%5Ccolor%7Bred%7D%7B%5Csqrt%7B6%7D%5Cmathrm%7B%5Cpi%7D%7D,選D.

幾何沒方向,快去請向量(2019課標(biāo)Ⅰ,12)的評論 (共 條)

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