专题5.2平面向量基本定理及坐标表示六大题型（原卷版）_11.4-2_2026-11-04

      
      
专题5.2 平面向量基本定理及坐标表示【六大题型】

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D¢ 3
Ġ>İ
9¢\2024·XY·ġģĤĥ]Ēž !𝑎= (1,2),𝑏= (−2,1).²(𝑥𝑎−𝑏)//(𝑎−𝑥𝑏)_”𝑥= \    ]
A¢−1
B¢0
C¢1
D¢2
10¢\2023·ŠĝſŽ·‚ġ]£ß¸5_ÁÂw𝐴𝐵𝐶𝐷ݎƀŪw_𝐶𝐷∥𝐴𝐵_𝐶𝐷=
1
2𝐴𝐵_𝐸_𝐹EåÝ𝐷𝐶_𝐴𝐸
%1_²𝐴𝐷= 𝜆𝐴𝐵+𝜇𝐵𝐹(𝜆,𝜇∈𝑅)_”\    ]
  
      
      
A¢𝜆=
7
2
B¢𝜇= 2
C¢𝜆=
7
4
D¢𝜇= 1
11¢\23-24 ‚c·ĜňƁ·ųēŴ‘]£ß_Mâw𝐴𝐵𝐶𝐷_𝐸Ý𝐴𝐵1_𝑀Ý/ē𝐴𝐷c%Ũ1_𝐵𝑀= 𝜆
𝐵𝐸+𝜇𝐵𝐷_”¾È†ÐMé%e\    ]
A¢þ𝑀Ý/ē𝐴𝐷c%1€_𝜆+ 𝜇=
3
2
B¢𝜆𝜇%8Œ9Ý
1
2
C¢𝜇%æ9;<Ý[0,1]
D¢𝜆+ 𝜇%æ9;<Ý 1
2 ,2
‚>uv
12¢\2024·cd·ġģĤĥ]£ß_ņw𝐴𝐵𝐶𝐷_𝐸Ý𝐵𝐶1_𝐴𝐸S𝐵𝐷Nj1𝐹_²¼𝐴𝐵= 𝑎_𝐴𝐷= 𝑏Þ
Ý !%¦¨"_” !𝐴𝐹»45Ý      
\𝑎、𝑏45].
  
13¢\2024·Ňňňʼn·Ġġ] !𝐴𝐵= (−3,6)_𝐴𝐶= (𝑚,5)_𝐶𝐷= (−1,4)¢²𝐴,𝐵,𝐷‚1./_”𝑚=
      
¢
14¢\2024·ĭĮĖƂ·¦ġ]£ß_ÁÂw𝐴𝐵𝐶𝐷eÂĩÝ 1 %Mâw_Ĩĩ CD ƃ E_±ç𝐷𝐸＝2𝐶𝐷¢Ũ
1 P l1 A ˆƄ_ƅMâw%ÂƆƇ€ƈâ ,Ũ¦ƉƊČ A 1_𝐴𝑃= 𝜆𝐴𝐵+𝜇𝐴𝐸_”𝜆+ 𝜇%æ9;<
Ý      
¢
      
      
Á>6Ƌ
15¢\23-24 ¦¾·Šňƌŋ·ųēŴ‘]Ēž𝐴(0,1)_𝐵(3,2)_𝐶(−1,5)¢
(1)²𝐴𝐵−2𝐴𝐶= (𝑚,𝑛)_(𝑚_𝑛~
(2)²𝐴𝐷= 2𝐴𝐵+4𝐴𝐶_(1𝐷%*+¢
16¢\23-24 ¦¾·ƍŝŐƎ·Ťť]£ß_ŠŽƀŪw𝐴𝐵𝐶𝐷_2𝐴𝐷= 2𝐷𝐶= 2𝐶𝐵= 𝐴𝐵= 2𝑎_𝐸_𝐹Eå
Ý𝐴𝐵_𝐴𝐷%1_𝐵𝐹S𝐷𝐸Nj1𝑀.
(1)𝐴𝐷_𝐴𝐸45𝐵𝐹~
(2)(/ē𝐴𝑀%ĩ.
17¢\23-24 ¦¾·ƏƐƑŽ·Ť] !𝑎= (3,2)_𝑏= (−1,2)_𝑐= (4,1)¢
(1)(Ħħ𝑎= 𝑚𝑏+𝑛𝑐%°
 m_n~
(2)² 𝑎+ 𝑘𝑐// 2𝑏−𝑎_(°
 k¢
      
      
18¢\23-24 ¦¾·ʼnŲƒƓ·Ťť]£ß_Š▱𝐴𝐵𝐶𝐷_E_H Eåe AD_BC %1_𝐴𝐹= 2𝐹𝐵_G
Ý DF S BE %N1¢
(1)˜ !𝐴𝐵= 𝑎_𝐴𝐷= 𝑏_Ɣs !𝑎_𝑏Ý"¹45𝐵𝐸_𝐷𝐹~
(2)²𝐴𝐶= 𝑚𝐵𝐸+𝑛𝐷𝐹_( m_n %9~
(3)(ƕ
A_G_H ‚1./¢
19¢\23-24 ¦¾·ŇňŦŧ·ųēŴ‘]Ēž𝑒1,𝑒2eg§¨©./%śŜ !_𝐴𝐵= 2𝑒1 + 𝑒2,𝐵𝐸= −
𝑒1 +𝜆𝑒2,𝐸𝐶= −2𝑒1 + 𝑒2_®𝐴,𝐸,𝐶‚1./¢
(1)(°
𝜆%9~
(2)²𝑒1 = (2,1),𝑒2 = (2,−2)_(𝐵𝐶%*+~
(3)Ēž𝐷(3,5)_Š\2]%UV¾_²𝐴,𝐵,𝐶,𝐷Á1ƆƖ€ƈƖêŸÑÁÂw_(1𝐴%*+¢