1、空間與平面的比較

лᚡȈޫ໣྆܉

1ˣ䨢攻冯⸛朊䘬㭼庫

(1)⸛朊⎗ ↢ 㡅Ḻ䚠✪䚜ᷳ䚜䶂ˤ

(2)䨢攻⎗ ↢ 㡅Ḻ䚠✪䚜ᷳ䚜䶂ˤ

2ˣ㰢⭂ᶨ䚜䶂 (1)䚠䔘ℑ溆

(2)ᶨ溆冯䚜䶂㕡⎹

3ˣ㰢⭂ᶨ⸛朊 (1)ᶵℙ䶂ᶱ溆

(2)ᶨ䚜䶂冯䶂⢾ᶨ溆

(3)Ḵ䚠Ṍ䚜䶂炷Ṍ㕤ᶨ溆炸 (4)Ḵ⸛埴䚜䶂

4ˣ䚜䶂冯䚜䶂䘬斄Ὢ (1) (2)

(3) (4)

­ ­

° ®

°° ¯

®° ­

° ®

° ¯

¯

䚠Ṍ Ṍ㕤ᶨ溆

慵⎰

ᶵ䚠Ṍ ⸛埴

㬒㕄炷䨢攻ㇵ㚱炸 ㆹ

5ˣ⸛朊冯⸛朊䘬斄Ὢ (1)⸛埴

(2)慵⎰

(3)䚠Ṍ㕤ᶨ䚜䶂 6ˣ䚜䶂冯⸛朊䘬斄Ὢ

(1)⸛埴

(2)䚜䶂句⛐⸛朊ᶲ (3)Ṍ㕤ᶨ溆

7ˣ✪䚜䘬⿏岒

䨢攻ᷕ炻䚜䶂冯⸛朊㚱ẍᶳ䘬✪䚜⿏岒烉

(1)劍ᶨ䚜䶂L冯⸛朊E䚠Ṍ㕤A炻侴L冯⛐Eᶲᶼ忶A溆ᷳ㭷ᶨ䚜䶂✪

䚜炻⇯䧙L冯E✪䚜ˤ炷䚜䶂冯⸛朊✪䚜⭂佑炸

(2)劍ᶨ䚜䶂L冯⸛朊E䚠Ṍ㕤A炻侴L冯⛐Eᶲᶼ忶A溆ᷳ䚠䔘ℑ䚜䶂✪

䚜炻⇯L冯E✪䚜ˤ炷䚜䶂冯⸛朊✪䚜⭂䎮炸

ἳ柴1烉

ᶳ↿㔀徘⒒ṃ㗗㬋䡢䘬烎

(1)⛐⸛朊ᶲ炻劍ℑ䚠䔘䚜䶂ᶵ䚠Ṍ炻⇯⬫Ᾱ⽭⸛埴 (2)⛐䨢攻ᷕ炻劍ℑ䚠䔘䚜䶂ᶵ䚠Ṍ炻⇯⬫Ᾱ⽭⸛埴

(3)⛐⸛朊ᶲ炻ảシℑ䚠䔘䚜䶂ᶨ⭂㚱℔✪䶂炷ṵ⛐娚⸛朊ᶲ炸 (4)⛐䨢攻ᷕ炻ảシℑ䚠䔘䚜䶂ᶨ⭂㚱℔✪䶂

(5)⛐䨢攻ᷕ炻䚠Ṍ䘬ℑ䚠䔘⸛朊ᶨ⭂㚱℔✪朊炷℔✪朊㗗㊯冯娚ℑ⸛朊悥✪䚜 䘬⸛朊炸

Sol烉

ἳ柴2烉

ᶳ↿㚱斄䨢攻䘬㔀徘炻⒒ṃ㗗㬋䡢䘬烎

(1)忶⶚䞍䚜䶂⢾ᶨ溆炻ˬ〘㚱˭ᶨ⸛朊冯㬌䚜䶂✪䚜

(2)忶⶚䞍䚜䶂⢾ᶨ溆炻ˬ〘㚱˭ᶨ⸛朊冯㬌䚜䶂⸛埴

(3)忶⶚䞍⸛朊⢾ᶨ溆炻ˬ〘㚱˭ᶨ䚜䶂冯㬌⸛朊⸛埴

ἳ柴3烉

⎛⚾䁢ᶨ㬋䩳㕡橼炻娎⓷ᶳ↿ỽ侭㬋䡢烎 (1)EA EG

K K

˜ 0

(2)ED EF

K K

˜ 0 (3)EF EH

K K K

AC (4)EC AG

K K

˜ 0

(5)EF EA EH

K K K K

EC Sol烉

ἳ柴4烉

⤪ᶲ⚾炻ᶳ↿ỽ侭冯AFℙ⸛朊烎

(1)FC (2)EG (3)HB (4)HD (5)GD Sol烉

ἳ柴5烉

ᶨᾳ㬋䩳㕡橼䘬ℓᾳ枪溆ᷕ炻⶚䞍㚱⚃ᾳ枪溆炻⼤㬌ᷳ攻䘬嶅暊悥㗗1炻㯪 㬌㬋䩳㕡橼䘬橼䧵烎

Sol烉

A B

C D

E F

G H

лᚡȈή࠮ጤۢ౪

䚜䶂L⛐⸛朊Eᶲ炻溆C⛐Lᶲ炻溆B⛐EᶲỮᶵ⛐Lᶲ炻溆Aᶵ⛐Eᶲ炻

⇯烉

(1)劍ABAEᶼBCAL Ÿ (2)劍ABAEᶼACALŸ

(3)ϛԚҴȈACALᶼBCAL Ÿ AB✪䚜E Pf烉

ἳ柴1烉

姕A B C, , ᶱ溆⛐⸛朊Eᶲ炻ᶼABABCˣPAAE㕤溆Aˤ⶚䞍

3 , 4 , 12

PA AB BC 炻㯪PCˤ

Sol烉

L

A

C

B D

P

ἳ柴2烉

ᶵℙ䶂ᶱ⮬䶂OX OY OZJJJK JJJK JJJK, ,

Ḻㆸ30⹎炻P溆⛐OXJJJK

ᶲ炻ᶼOP 2ˤP军⸛

朊OYZ䘬㈽⼙䁢Q炻䓙Q军OYJJJK

ᷳ✪嵛䁢Rˤ⍰䚜䶂QRHJJG ṌOZHJJG

㕤S溆ˤ娎㯪烉 (1)OR攟 (2)RS攟 (3)PS²PR²

Sol烉

ἳ柴3烉

䚜奺ːABCᷕ炻ɳC䁢䚜奺ˤAC烌15炻BC烌20炻冒C溆ἄ⸛朊ABCᷳ✪

䚜䶂㭝PC炻⶚䞍PC烌5炻㯪P溆⇘㕄怲AB䘬✪䚜嶅暊ˤ Sol烉

O

Y

Z X

P

S

R Q

лᚡȈڎ८ِ

˰⭂佑˱

⛐⸛朊E冯FᷳṌ䶂炷䧄炸PQHJJG

ᶲ⍾ᶨ溆A炻⛐EᶲἄHJJGABAPQHJJG

ˤ⛐Fᶲ ἄHJJGACAHJJGPQ

炻⇯ ˤ

˰㯪㱽˱

(1)检⻎⭂䎮炷䞍ᶱ怲炸ŸcosT

(2)䓐⎹慷炷⹏㧁⊾炸ŸcosT

ἳ柴1烉

姕A B C, , ᶱ溆⛐⸛朊Eᶲ炻ᶼABABCˣPAAE㕤溆Aˤ⶚䞍

3 , 4 , 12

PA AB BC 炻㯪PCˤ

Sol烉 A

B

C

ἳ柴2烉

ℑ⸛朊E F, Ṍ㕤ᶨ䚜䶂CDˤA㗗⸛朊Eᶲᶨ溆炻ᶼA⛐F䘬㬋⮬⼙䁢Bˤ

⶚䞍‘ACB䁢30⹎炻E F, ㇨䁢ㆸ䘬ℑ朊奺䁢45⹎炻㯪cos‘ACDˤ Sol烉

ἳ柴3烉

⚃朊橼ABCDᷕ炻ẌAB AC AD 4炻BC CD DE 2炻劍T䁢⸛朊 ABC␴⸛朊BCD㇨⣦ᷳḴ朊奺炻㯪烉

(1)cosT (2)⚃朊橼橼䧵

Sol烉 橼䧵℔⺷烉

㞙橼橼䧵烌⸽朊䧵 ǘ 檀 抸橼橼䧵烌1

3 ǘ ⸽朊䧵 ǘ 檀

ἳ柴4烉

⤪ᶳ⚾炻⮯ᶨ⻝㬋㕡⼊䘬䳁ABCD㱧叿⮵奺䶂BD㐢崟炻ἧ⼿ɳABC = 60q炻 㯪Ḵ⸛朊ABD冯BCD䘬⣦奺ˤ

Sol烉

ἳ柴5烉

㚱ᶨ⎬䧄攟⛯䁢8䘬慹⫿⟼⼊炻℞“朊䁢⚃ᾳ䫱儘ᶱ奺⼊炻⸽朊ᷳ怲攟䁢6 䘬㬋㕡⼊炻劍䚠惘ℑ“ᷳ⣦奺䁢D炻⸽朊冯“朊ᷳ⣦奺䁢E炻娎㯪烉

(1)cosD (2) cosE (3)㬌Ḽ朊橼橼䧵 Sol烉

栆柴1烉

ᶳ↿㔀徘ỽ侭㬋䡢烎

(1)䨢攻ᷕ悐䚠Ṍ䘬ℑ䚜䶂⽭㚫⸛埴 (2)✪䚜㕤⎴ᶨ⸛朊䘬ℑ䚜䶂⽭Ḻ䚠⸛埴

(3)劍䨢攻䚜䶂L冯⸛朊EṌ㕤ᶨ溆炻⇯⬀⛐ⓗᶨ⸛朊⊭⏓Lᶼ␴E✪䚜

(4)劍䨢攻䚜䶂L冯⸛朊EḺ䚠✪䚜炻⇯⊭⏓L䘬⸛朊⽭冯E✪䚜

(5)䴎䨢攻ᷕℑ䚠䔘䚜䶂炻⇯⽭⬀⛐䚜䶂冯㬌ℑ䚜䶂⛯✪䚜 Ans烉245

栆柴2烉

⤪⚾炻OAʄ⸛朊E炻ᶼABAL炻⶚䞍OA烌3炻AB烌4炻BC烌3炻㯪OC攟ˤ

Ans烉 34

栆柴3烉

⤪⚾炻ℑ⋲⸛朊E F, 炻Ṍ㕤ᶨ䚜䶂OT炻A㗗⸛朊Eᶲᶨ溆炻ẌA⛐⸛朊F

ᷳ㬋⮬⼙䁢B炻⶚䞍ɳAOB䁢45⹎炻E F, ㇨⣦ᷳḴ朊奺⹎慷䁢60⹎炻姕 AOT T

‘ 炻㯪sinTῤˤ

Ans烉 3 6

栆柴4烉

⎛⚾䁢ᶨ㬋䩳㕡橼炻塓ᶨ⸛朊㇒↢ᶨᾳ⚃怲⼊ABCD炻℞ᷕB D, ↮⇍䁢䧄䘬

ᷕ溆炻ᶼEA烉AF 烌1烉2炻㯪cosɳDABˤ Ans烉 1

37

栆柴5烉

姕Ḵ⸛朊E F, Ṍ㕤ᶨ䚜䶂L炻⸛朊Eᶲ㚱ᶨ溆A炻A⛐F䘬㬋⮬⼙溆䁢B炻 冒BἄL䘬✪䶂炻✪嵛溆䁢Cˤ劍AB烌6炻AC烌12炻娎㯪烉

(1)BC攟 (2)ℑ⸛朊ᷳ扛⣦奺 Ans烉(1) 6 3 (2)30˚

栆柴6烉

ᶳ⚾㬋⚃朊橼ABCDᷕ炻劍⛐AB AC AD, , ᶲ↮⇍⍾溆P Q R, , ˤ⶚䞍AD✪ 䚜⸛朊PQRᶼAP烌6炻㯪

(1)AR (2)ːPQR䘬朊䧵

Ans烉(1)3 (2)9 2

лᚡȈҔѳ८ᡞޠܓ፵

˰⭂佑˱

䓙⚃ᾳ㬋ᶱ奺⼊㇨䳬ㆸ䘬䩳橼⚾⼊炻䧙䁢㬋⚃朊橼炻姕怲攟䁢aˤ

˰⿏岒1˱㬋⚃朊橼D ABC 炻D溆⮵'ABCᷳ㈽⼙溆Q䁢'ABCᷳ ˤ

˰⿏岒2˱㬋⚃朊橼D ABC 炻℞檀䁢 烊℞橼䧵䁢

Pf烉

˰⿏岒3˱劍㬋⚃朊橼D ABC ᷳℑ朊奺䁢T炻⇯cosT

˰⿏岒4˱㬋⚃朊橼D ABC 炻㬒㕄ℑ䧄炷㚱ᶱ䳬炸ᷳ嶅暊d =

Pf烉 A

C B D

˰⿏岒5˱㬋⚃朊橼D ABC 炻⢾㍍䎫ᷳ⋲⼹R

˰⿏岒6˱㬋⚃朊橼D ABC 炻ℏ↯䎫ᷳ⋲⼹r

Pf烉

ἳ柴1烉

⶚䞍㬋⚃朊橼D ABC 䘬䧄攟䁢6炻⚆䫼ᶳ↿⓷柴烉 (1)檀 (2)橼䧵 (3)ℏ↯䎫⋲⼹

(4)⢾㍍䎫⋲⼹ (5)ℑ㬒㕄䧄嶅暊 (6)姕T䁢ℑ朊奺炻㯪sinT Sol烉

ἳ柴2烉

姕ABCD䁢ᶨ⚃朊橼炻侴AB AC AD 1炻ɳDAB ɳDAC ɳBAC 30q炻 㯪ːBCD䘬朊䧵ˤ

Sol烉

ἳ柴3烉

怲攟䁢2䘬㬋䩳㕡橼䘬ℓᾳ枪溆炻劍怠⍾ᶱᾳ枪溆忋ㆸ㬋ᶱ奺⼊炻㯪㬌㬋ᶱ 奺⼊朊䧵ˤ

Sol烉

栆柴1烉

⤪⚾炻劍D ABC 䁢ᶨ㬋⚃朊橼炻怲攟䁢10炻DH✪䚜⸛朊ABC㕤H炻⇯

ᶳ↿ỽ侭㬋䡢烎 (1)H䁢ːABCᷳℏ⽫ġ (2)BD

K K

AAC

ġ (3)DH

3 3 10 ġ

(4)劍⸛朊ABC冯⸛朊ADC䘬⣦奺䁢T炻⇯cosT 3 1ġ

(5)AD冯BC䘬嶅暊䁢5 2 Ans烉125

栆柴2烉

⤪ᶳ⚾炻㬋㕡⼊ABCD䘬怲攟䁢a炻侴P炻Q⎬䁢BC炻CD䘬ᷕ溆炻Ṳ⮯㬌 㬋㕡⼊㱧嘃䶂⎹ᶲ㐢崟炻ἧB炻C炻Dᶱ溆慵⎰炻Ẍ㬌慵⎰溆䁢R炻㯪⚃朊橼A-PQR

ᷳ橼䧵ˤ

Ans烉24 a3

栆柴3烉

攟㕡橼䘬ᶨ枪溆O炻ẍO䁢枪溆䘬ᶱ怲䁢OA炻OB炻OC炻劍AB= 3炻AC

= 2炻‘BAC = 60q炻娎㯪

(1)OA² (2)OB² (3)OC² (4)O⇘⸛朊ABC䘬嶅暊 Ans烉(1) 3ġ (2) 6ġ (3) 1ġ (4)

3 6

栆柴4烉

ᶳ⚾㗗ᶨᾳ㬋⚃奺抸炻⬫䘬⸽朊㗗ᶨᾳ怲攟䁢2䘬㬋㕡⼊炻㬌㬋⚃奺抸䘬檀䁢1炻 㯪ℑ䚠惘“朊䘬⣦奺ᷳ⹎㔠ˤ

x

y z

y

x z

лᚡȈޫ໣ֳዀق

1ˣ䨢攻⛸㧁䲣烉

⛐䨢攻ᷕả⍾ᶨ溆O炻忶O溆ἄℑℑḺ䚠✪䚜䘬ᶱ㡅䚜䶂炻⛐忁ᶱ㡅 䚜䶂ᶲ炻⎬⍾ᶨᾳ㕡⎹ 䁢㬋㕡⎹炻᷎⍾怑䔞攟⹎ 䁢╖ỵ攟炻忁㧋㭷㡅 䚜䶂⯙嬲ㆸẍO溆䁢⍇溆䘬㔠䶂炻↮⇍䁢x庠ˣy庠ˣz庠炻䴙䧙䁢⛸㧁庠ˤ ㆹᾹ䧙⍇溆Oˣx庠ˣy庠冯z庠䳬ㆸḮ䨢攻⛸㧁䲣ˤ

(1)⎛ㇳ䲣䨢攻⛸㧁 (2)ⶎㇳ䲣䨢攻⛸㧁

2ˣ⛸㧁⸛朊烉

⛐䨢攻⛸㧁ᷕ

(1)x庠冯y庠㇨⛐䘬⸛朊䧙䁢xy⸛朊 (2)y庠冯z庠㇨⛐䘬⸛朊䧙䁢yz⸛朊 (3)z庠冯x庠㇨⛐䘬⸛朊䧙䁢zx⸛朊

3ˣ⌎旸烉

ᶱᾳ⛸㧁⸛朊炻⮯䨢攻↮ㆸ8ᾳ悐↮炻㭷ᶨᾳ悐↮䧙䁢ᶨᾳ⌎旸ˤ (1)䫔ᶨ⌎旸䁢

^

^{( , , ) |x y z x!0,y!0,z!0}

`

(2)℞检⌎旸ᷳ枮⸷㛒㖶䡢夷⭂

лᚡȈֳዀߓұݳ

姕P䁢䨢攻ᷕᶨ溆炻忶P溆↮⇍ἄᶨ⸛朊冯x庠ˣy庠ˣz庠✪䚜炻⇯忁 ᶱ⸛朊枮㫉冯x庠ˣy庠ˣz庠Ṍ溆䘬⛸㧁a b c, , ↮⇍䧙䁢P溆䘬x⛸㧁ˣy⛸ 㧁冯z⛸㧁ˤẍP a b c( , , )堐䣢P溆䘬⛸㧁䁢( , , )a b c ˤ

˰⿏岒1˱⛸㧁庠ᶲ䘬溆

(1)x庠ᶲ䘬溆⛸㧁⽭✳⤪Ÿ Ÿ (2)y庠ᶲ䘬溆⛸㧁⽭✳⤪Ÿ Ÿ (3)z庠ᶲ䘬溆⛸㧁⽭✳⤪Ÿ Ÿ

˰⿏岒2˱⛸㧁⸛朊ᶲ䘬溆⛸㧁

(1)xy⸛朊ᶲ䘬溆⛸㧁⽭✳⤪Ÿ Ÿ (2)yz⸛朊ᶲ䘬溆⛸㧁⽭✳⤪Ÿ Ÿ

O

x

z

y ( , , ) P a b c

ἳ柴1烉

⤪⚾炻娎㈦↢⎬溆⛸㧁⍲P a b c( , , )⮵⎬庠⍲⎬朊䘬㈽⼙溆⍲⮵䧙溆⹏㧁ˤ

溆 A B C Q R S

⛸㧁

㬋⮬⼙⛸㧁 ⮵䧙溆⛸㧁 ⇘庠(朊)嶅暊 x庠

y庠 z庠 xy⸛朊 yz⸛朊 zx⸛朊

⍇溆

x

z

O y

B A

S

C R

P

Q

лᚡȈڎᘉ຾ᚕϵԓІϸᘉϵԓ

1ˣℑ溆嶅暊℔⺷

(1)⸛朊ᶲℑ溆A x y( , )_{1 1} ˣB x y( , )_{2 2} Ÿd A B( , ) AB (2)䨢攻ᷕℑ溆A x y z( , , )_{1 1 1} ˣB x y z( , , )_{2 2 2} Ÿd A B( , ) AB 2ˣ↮溆℔⺷

(1)⸛朊ᶲℑ溆A x y( , )_{1 1} ˣB x y( , )_{2 2} 炻㺧嵛P⛐ABᶲᶼAP PB m n: :

ŸP溆⛸㧁䁢

(2)䨢攻ᷕℑ溆A x y z( , , )_{1 1 1} ˣB x y z( , , )_{2 2 2} 炻㺧嵛P⛐ABᶲᶼAP PB m n: :

ŸP溆⛸㧁䁢

ἳ柴1烉

⤪⚾炻⶚䞍F(1, 2,3)炻娎㯪℞检⎬溆⛸㧁⍲F溆⇘⎬嗽䘬嶅暊ˤ

x

z

D y

C A

E

H _G

F

B

A溆 B溆 C溆 D溆 E溆 G溆 H溆

ἳ柴2烉

姕P溆⛐䫔ᶨ⌎旸炻ᶼP溆⇘x庠ˣy庠ˣz庠䘬嶅暊↮⇍䁢5ˣ 34ˣ 41炻 㯪P溆⛸㧁ˤ

Sol烉

ἳ柴3烉

⶚䞍P(2, 3,1) 炻娎㯪

(1)P⮵xy⸛朊䘬㬋⮬⼙ (2)P⮵yz⸛朊䘬⮵䧙溆 (3)P⮵y庠䘬⮵䧙溆 Sol烉

ἳ柴4烉

ᶨ≽溆P冯⍇溆䘬嶅暊䁢℞冯⎎ᶨ溆(2, 0, 0)嶅暊䘬ᶨ⋲炻娎㯪㺧嵛㬌㡅ẞ

ᷳ≽溆P㇨⼊ㆸ䘬年嶉⚾⼊ᷳ㕡䦳⺷ˤ Sol烉

ἳ柴5烉

⶚䞍A(1, 2,3)ˣB(2, 1, 1)炻㯪㺧嵛PA PBᷳ溆P㇨⼊ㆸ䘬年嶉㕡䦳⺷ˤ Sol烉

≽溆年嶉㕡䦳⺷

(1)姕≽溆P x y z( , , )

(2)ὅ≽溆㺧嵛ᷳ㡅ẞ↿⺷

(3)㯪≽溆Pᷳx y z, , 䘬斄Ὢ⺷

ἳ柴6烉

姕A B, ℑ溆⛸㧁↮⇍䁢A(2, 1, 2) ˣB( 1,5, ) z ᶼAB 7炻㯪z䘬⛸㧁ˤ Sol烉

ἳ柴9烉

姕P⛐xy⸛朊ᶲ炻ᶼ冯ᶱ溆A( 4,8, 2) ˣB(2,5, 6)ˣC(2, 0, 0)䫱嶅炻㯪Pῤˤ Sol烉

ἳ柴10烉

⶚䞍ᶨ㬋⚃朊橼℞ᷕᶱ枪溆(0, 0, 0)ˣ(2, 0, 0)ˣ(1,1, 2)炻㯪⎎ᶨ枪溆⛸㧁ˤ Sol烉

ἳ柴11烉

䨢攻ᷕᶨ溆P军ᶱ庠ᷳ嶅暊↮⇍䁢2,3, 4炻娎㯪溆P军⍇溆䘬嶅暊ˤ Sol烉

ἳ柴12烉

⤪ᶳ⚾炻㚱ᶨ怲攟䁢1䘬㬋㕡橼ˤṲ伖枪溆A㕤䨢攻⛸㧁䲣ᷕᷳ⍇溆(0, 0, 0)炻 伖枪溆B㕤㬋z庠ᶲ炻㯪枪溆Cᷳz⛸㧁

Sol烉

ἳ柴13烉

姕A(2, 6,3)ˣB(2, 2, 1) 炻⶚䞍C⛐HJJGAB

ᶲᶼ㺧嵛AC 3BC炻㯪C溆⛸㧁ˤ Sol烉

栆柴1烉

ᶨ䶂㭝AB⛐xy⸛朊炻yz⸛朊炻zx⸛朊ᶲ䘬㬋⮬⼙攟↮⇍䁢4炻 15炻 21炻 㯪AB攟ˤ

Ans烉 26

栆柴2烉

㬋⚃朊橼ABCD炻⶚䞍B C D, , 䘬⛸㧁↮⇍䁢B(0, 0, 0)ˣC(1, 0, 0)ˣD x y( , , 0)炻

℞ᷕx炻y䘮䁢㬋炻㯪

(1)D䘬⛸㧁 (2) A䘬⛸㧁 (3)姕A⛐⸽朊BCD㬋⮬⼙䁢H炻⇯H䘬⛸㧁 Ans烉(1) (

2 1,

2

3,0)ġ (2) ( 2 1,

6 3,

3 6 )ㆾ(

2 1,

6 3,

3

6 )ġ (3) ( 2 1,

6 3,0)

栆柴3烉

攟㕡橼ABCD-EFGH炷⤪ᶳ⚾炸ᷕ炻AB 1炻AE 2炻AD 3炻PA 2炻FQ 1炻㯪PQ䘬攟ˤ

Ans烉 6

栆柴4烉

姕䶂㭝ABᷳ攟䁢5炻㬌䶂㭝⛐xy⸛朊炻yz⸛朊ᶲᷳ㬋⮬⼙攟↮⇍䁢 19炻 21炻㯪㬌䶂㭝⛐zx⸛朊ᶲᷳ㬋⮬⼙攟ˤ

Ans烉 10

栆柴5烉

䨢攻ᷕℑ溆A(2, 1, 0) ˣC(1,1,1)ˤ⶚䞍B⛐ACᶲᶼAB BC: 1: 2炻㯪B溆

лᚡȈޫ໣Ӫ໕

1ˣ䨢攻⎹慷

⸦ỽ悐↮炻䨢攻冯⸛朊忳䬿⿏岒䚠⎴ˤẋ㔠悐↮炷⛸㧁⊾炸炻䨢攻⎹慷 㭼⸛朊⎹慷⣂ᾳ ˤ

姕A x y z( , , )_{1 1 1} ˣB x y z( , , )_{2 2 2} ˣ

K

a ( , , )a a a_{1 2 3}

ˣ

K

b ( , , )b b b_{1 2 3} 炻⇯

(1) AB

K

(2) |AB

K

|

(3) 䚠䫱烉劍

K K

a b Ÿ (4) ≈㱽烉

K K

a b (5) Ὢ㔠䧵烉r a

K

(6) ↮⎹慷烉劍 Ÿ

_______________炷O䁢ảシ溆炸 (7) ℏ䧵烉

K K

a b˜

(8) ⸛埴烉劍

K K

a // b Ÿ

Ÿ (9) ✪䚜烉劍

K K

a A b Ÿ

Ÿ (10)㞗大ᶵ䫱⺷Ÿ

(11)⣦奺烉cosT Ÿ

(12)

K

a

⛐

K

b

䘬㬋⮬⼙烌 Ÿ

(13)'ABC䘬朊䧵烌 (14)

K

a

䘬╖ỵ⎹慷烌 Ÿ

A P B

m n

ἳ柴1烉

⶚䞍

K

a (1, 2, 2)

ˣ

K

b (1, 1,1)

炻㯪(

K K K K

a2 b) (˜ a b) ˤ Sol烉

ἳ柴2烉

⶚䞍'ABC炻A(1,1,1)ˣB(0,3,3)ˣC(3,0, 2)炻㯪 (1)sinA (2)'ABC朊䧵 (3)AB

K

⛐AC

K

䘬㬋⮬⼙

Sol烉

ἳ柴3烉 (1, 2,3)

A ˣB(2,5,3)ˣC(2, 6, 4)炻⶚䞍D冯㬌ᶱ溆㥳ㆸᶨ⸛埴⚃怲⼊炻㯪D溆

⛸㧁ˤ Sol烉

ἳ柴4烉

(4,1,1)

A B(0, 6, 0) C( 1,1, 2) BA BC

K K

˜

ἳ柴5烉

K

u (3, 2, 4)

ˣ

K

v (2,1, 1)

炻姕

K K K

w u t v

炻㯪䔞t䁢⣂⮹㗪炻

K

w

䘬攟⹎

㚨⮷ˤ Sol烉

ἳ柴6烉

⶚䞍'ABCᷕ炻AB

K

(1, 2, 2)

ˣAC

K

(0,3, 4)

炻娎㯪'ABC䘬␐攟ˤ Sol烉

ἳ柴7烉

姕

K

a (1, 2, 2)

炻㯪

K

a

⎴⎹ᷳ╖ỵ⎹慷ˤ Sol烉

ἳ柴8烉

姕A(1, 0, 1) ˣB(2, 2,1) ˣC(3, 2,1)炻⶚䞍AB CD

K K

//

ᶼ |CD

K

| 6

炻娎㯪D溆

⛸㧁ˤ Sol烉

ἳ柴9烉

姕

K

a ( , 4,3x z)

ˣ

K

b ( , 2,1)y

ˣ

K

c (1, 4 ,z x )

ᶼ

K K

a // b

ˣ

K K

a A c 炻 㯪( , , )x y z ˤ

Sol烉

ἳ柴10烉

䨢攻ᷕ⚃溆A B C D炻劍, , , AB 1ˣBC 2ˣCD 3ᶼ‘ABC ‘BCD 120q炻 AB

K

冯CD

K

⣦奺䁢60q炻㯪AD攟ˤ Sol烉

ἳ柴11烉

姕X Y Z, , ᶱ溆䁢⇍句㕤䨢攻⛸㧁x y z, , 庠ᶲ炻㯪‘XOY⛐xy⸛朊ᶲᷳ奺⸛↮

䶂冯‘YOZ⛐yz⸛朊ᶲᷳ奺⸛↮䶂䘬⣦奺ˤ Sol烉

ἳ柴12烉

⤪ᶳ⚾炻攟㕡橼ᷳ攟炻⮔炻檀⎬䁢4炻5炻3炻娎㯪AG

K

冯FD

K

䘬⣦奺ˤ Sol烉

ἳ柴13烉

⤪ᶳ⚾炻ABCD䁢㬋䩳㕡橼䘬ᶨᾳ朊炻PˣQ↮⇍䁢BCˣCD䘬ᷕ溆炻O䁢 㬋䩳㕡橼䘬ᷕ⽫炻㯪cos (ɳPOQ)ˤ

Sol烉

ἳ柴14烉

⤪ᶳ⚾炻攟㕡橼ABCD-EFGHᷕ炻AB 4炻BC 2炻AE 3炻㯪AG CH

K K

˜ ˤ Sol烉

栆柴1烉

姕

K

a _{(x炻y炻z)炻}

K

b (2炻3炻6)炻劍 |

K

a _{| 5炻㯪2x 3y 6z}

䘬㚨⣏ῤˤ Ans烉35

栆柴2烉

⶚䞍⸛埴⚃怲⼊ABCDᷕ炻A(1炻2炻3)炻B(4炻5炻6)炻C( 5炻8炻7)炻 㯪D溆⛸㧁ˤ

Ans烉( 8炻5炻4)

栆柴3烉

姕

K

a (2炻 1炻 2)炻

K

b _{(1炻2炻2)炻}

c a t b

K K K

炻 (1)劍

K

c _A

K

a

炻⇯t ġ ġ ġ ġ ġ ˤ (2)劍

K

c

⸛↮

K

a ˣ

K

b

䘬⣦奺㗪炻t ġ ġ ġ ġ ġ ˤ Ans烉(1)9

4ġ (2) 1

栆柴4烉

姕a炻b炻c⛯䁢㬋㔠ᶼabc 9炻⇯

c b a

16 9

4 ᷳ㚨⮷ῤ䁢ġ ġ ġ ġ ġ ˤ

Ans烉9 栆柴5烉

ᶨ䧄攟䁢a䘬㬋⚃朊橼ABCD炻CDᷳᷕ溆䁢M炻BCᷳᷕ溆䁢N炻⇯

K K K K

лᚡȈϵ࠮Ӫ໕

1ˣ℔✪⎹慷䘬⭂佑

劍

K K

a A n

ᶼ

K K

b A n

炻⇯䧙

K

n

䁢

K K

a , b

䘬℔✪⎹慷ˤ Ɉ℔✪⎹慷ᶵⓗᶨ炻Ữ⛯

2ˣ℔✪⎹慷䘬忳䬿

劍⶚䞍

K

a ( , , )a a a_{1 2 3}

ˣ

K

b ( , , )b b b_{1 2 3} 炻

⇯℞℔✪⎹慷

K

n

ˤ

Pf烉

姕

K

n ( , , )x y z

炻恋湤

1 2 3

1 2 3

0 0

0 0

a x a y a z

a n a n

b x b y b z

b n b n

­ A ­ ˜ ­

° œ ° œ

® ® ®¯

° A ° ˜

¯ ¯

K K K K

K K K K

^{炷柴✳烉㯪㭼ἳ炸}

Ɉ姀㱽烉⍣柕⍣⯦㱽

1 1

a a

b b

K K

^ab₂^{2 a}b₃^{3 a}b₁^{1 a}b²₂ ^ab³₃

2 3 3 1 1 2

2 3 3 1 1 2

____________________________________

( a a , a a , a a )

n b b b b b b

K

Ḵ昶埴↿⺷ῤ䘬忳䬿 a b ad bc

c d

ἳ柴1烉

姕

K

a (1,1,1)

ˣ

K

b (5,3, 2)

炻㯪

K

a 冯

K

b

ᷳ℔✪⎹慷ᷳ╖ỵ⎹慷ˤ Sol烉

ἳ柴2烉

⶚䞍ᶨ⸛朊E r//

K

₁ (1,1, 1)

ᶼE r//

K

₂ (2, 1,3)

炻娎㯪⸛朊E䘬㱽⎹慷ˤ Sol烉

ἳ柴3烉

⶚䞍xyzz0ᶼ2x y z 0ˣx y z 0炻㯪x y z: : ˤ Sol烉

栆柴1烉

姕

K

a (2,3,1)

ˣ

K

b (1, 1,0)

炻㯪

K

a 冯

K

b

ᷳ℔✪⎹慷ᷳ╖ỵ⎹慷ˤ

Ans烉 1 1 5

( , , )

3 3 3 3 3 3 r

лᚡȈѵᑗȞ K a b u K _{࢑ ȟ}

1ˣ⢾䧵䘬⭂佑

⶚䞍

K

a ( , , )a a a_{1 2 3}

ˣ

K

b ( , , )b b b_{1 2 3}

炻⇯ℑ⎹慷䘬⢾䧵炷

K K

au b

炸ṵ㗗

⎹慷炻ᶼ㺧嵛ᶳ↿ℑ㡅ẞ (1)㕡⎹烉

K K K

au b // n

ᶼ㺧嵛⎛ㇳ夷⇯ 炷nK

堐䣢aK 冯bK

ᷳ℔✪⎹慷炸 (2)⣏⮷烉|

K K

au b |

K

a 冯

K

b

㇨⯽攳ᷳ⸛埴⚃怲⼊朊䧵ῤ

恋湤炻

K K

au b

˰婒㖶˱

2ˣ埴↿⺷冯朊䧵䘬斄Ὢ (1)⛐⸛朊ᶲ烉

1 2

( , ) a a a

K

ˣ

K

b ( , )b b_{1 2}

炻⇯

K

a 冯

K

b

㇨䳬ㆸᷳ

ᶱ奺⼊朊䧵䁢 ⸛埴⚃怲⼊朊䧵䁢

(2)⛐䨢攻ᷕ烉

1 2 3

( , , ) a a a a

K

ˣ

K

b ( , , )b b b_{1 2 3}

炻⇯

K

a 冯

K

b

㇨䳬ㆸᷳ

ᶱ奺⼊朊䧵䁢 ⸛埴⚃怲⼊朊䧵䁢

ἳ柴1烉

⸛朊ᶲ㚱ᶱ溆A(1, 2)ˣB( 1,3) ˣC(3, 7)炻㯪'ABCᷳ朊䧵ˤ Sol烉

ἳ柴2烉

䨢攻ᷕ㚱ᶱ溆A(1, 2,3)ˣB( 1,3, 2) ˣC(3,3,1)炻㯪'ABCᷳ朊䧵ˤ Sol烉

栆柴1烉

㯪

K

u (1, 2)

ˣ

K

v (3, 2)

㇨⚵ㆸ䘬⸛埴⚃怲⼊䘬朊䧵ˤ Ans烉8