在正三棱柱

中,所有棱的長(zhǎng)度都是2,

是

邊的中點(diǎn),問(wèn):在側(cè)棱

上是否存在點(diǎn)

,使得異面直線

和

所成的角等于

.
在側(cè)棱

上不存在點(diǎn)

,使得異面直線

和

所成的角等于
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以

點(diǎn)為原點(diǎn),建立如圖所示的空間直角坐標(biāo)系

.

因?yàn)樗欣忾L(zhǎng)都等于2,所以

.
假設(shè)在側(cè)棱

上存在點(diǎn)

,使得異面直線

與

所成的角等于

,
可設(shè)

,
則

.
于是,

.
因?yàn)楫惷嬷本€

和

所成的角等于
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,
所以

和

的夾角是

或

.
而

,
所以

,解得

,但由于

,
所以

點(diǎn)不在側(cè)棱

上,
即在側(cè)棱

上不存在點(diǎn)

,使得異面直線

和

所成的角等于

.
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在
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(1)證明:

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
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
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
,
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
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
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
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
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
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
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
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
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
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
與

的中點(diǎn),點(diǎn)

在平面

上的射影是

的重心

,求點(diǎn)

到平面

的距離.
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已知直線

平面
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,直線
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
平面

,給出下列命題,其中正確的是( )
①

②
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③

④
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若

,

,

是平面

內(nèi)的三點(diǎn),設(shè)平面

的法向量

,則

________________。
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