Question

分別求點(diǎn)P(a,b,c)關(guān)于坐標(biāo)平面、坐標(biāo)軸、坐標(biāo)原點(diǎn)的對(duì)稱(chēng)點(diǎn)的坐標(biāo).

Answer 1

思路分析:要認(rèn)真考慮各情況下的對(duì)稱(chēng)點(diǎn)應(yīng)具有的性質(zhì),進(jìn)而得出對(duì)稱(chēng)點(diǎn)與點(diǎn)P的關(guān)系.

解:點(diǎn)P(a,b,c)關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn)為P′(x,y,z),則PP′的中點(diǎn)為(0,0,0),故點(diǎn)P(a,b,c)關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn)為(-a,-b,-c).

圖4-3-1

    點(diǎn)P(a,b,c)關(guān)于xOy平面的對(duì)稱(chēng)點(diǎn)為P′,則直線PP′垂直于xOy平面,直線上的點(diǎn)的坐標(biāo)滿足x=a,y=b,且線段PP′的中點(diǎn)在xOy平面內(nèi)(z=0),故點(diǎn)P(a,b,c)關(guān)于xOy平面的對(duì)稱(chēng)點(diǎn)為(a,b,-c).類(lèi)似地,點(diǎn)P(a,b,c)關(guān)于yOz平面的對(duì)稱(chēng)點(diǎn)為(-a,b,c),關(guān)于zOx平面的對(duì)稱(chēng)點(diǎn)為(a,-b,c).

點(diǎn)P(a,b,c)關(guān)于y軸的對(duì)稱(chēng)點(diǎn)為P′,線段PP′被y軸垂直平分,過(guò)點(diǎn)P作與y軸垂直的平面,與y軸相交于Q(0,a,0),PQ⊥y軸,PP′中點(diǎn)為Q,故點(diǎn)P(a,b,c)關(guān)于y軸的對(duì)稱(chēng)點(diǎn)為P′(-a,b,-c).

類(lèi)似地,點(diǎn)P(a,b,c)關(guān)于x軸的對(duì)稱(chēng)點(diǎn)為P′(a,-b,-c),點(diǎn)P(a,b,c)關(guān)于z軸的對(duì)稱(chēng)點(diǎn)為P′(-a,-b,c).

  綠色通道:記憶方法:“關(guān)于誰(shuí)誰(shuí)不變,其余的則相反”,如:關(guān)于x軸對(duì)稱(chēng)的點(diǎn),橫坐標(biāo)不變,縱坐標(biāo)、豎坐標(biāo)變?yōu)樵瓉?lái)的相反數(shù);關(guān)于xOy坐標(biāo)平面對(duì)稱(chēng)的點(diǎn),橫、縱坐標(biāo)不變,豎坐標(biāo)變?yōu)樵瓉?lái)的相反數(shù).