专题5.1平面向量的概念及线性运算五大题型（原卷版）_11.4-2_2026-11-04

      
      
专题5.1 平面向量的概念及线性运算【五大题型】

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(1)
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7/.
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Lw
.
(4)
(λ,μ ˆ×
)LÖ A,B,C ·<7/Lw λ+μ=1.
*ØٌÚÛ
1~t<h" !gh
Ö P ˆ/Ü AB "t<LO ˆZݾ<Lw
.
2~
(λ,μ ˆ×
)LÖ A,B,C ·<7/Lw λ+μ=1.
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¾X”Þ=ß !"àáLâã="Xß !" ²äX
=åæ-… !"åç0L߅ !Xè†éêëì.
 1   !"#$%&
» 1V2024·QR·íîïðW½x !𝑎L𝑏ˆ„… !Lw“ !𝑎L𝑏"ñ¸ˆ 180°”X“𝑎//𝑏”"V    W
A~òB”ó=ëì
B~ó=”òBëì
C~ò=ëì
D~ô”òB†”ó=ëì
õh 1-1V2024·ö÷··íWÖ𝑎,𝑏ˆ„… !Lw“
𝑎
|𝑎| =
𝑏
|𝑏|”X“𝑎,𝑏7/”"V    W
A~ò=ëì
B~òB”ó=ëì
C~ó=”òBëì
D~ô”òB†”ó=ëì
õh 1-2V2023·øùúû··íW½x𝐴𝐵𝐶𝐷XµgLü𝑝
𝐴𝐵= 2𝐷𝐶L𝑞
𝐴𝐵𝐶𝐷XýgLw𝑝X𝑞
"ëìV    W
A~òB”ó=
B~ó=”òB
C~ò=
D~ô”òB†”ó=
õh 1-3V2024·þÿĀā·íîïðWĂă{µg𝐴𝐵𝐶𝐷"gĄÍÎąĆ"XV    W
A~Ö𝐴𝐷= 𝐵𝐶Lwµg𝐴𝐵𝐶𝐷ˆzµg
B~Ö𝐴𝐷=
1
3𝐵𝐶Lwµg𝐴𝐵𝐶𝐷ˆýg
C~Ö𝐴𝐵= 𝐷𝐶LÔ|𝐴𝐵| = |𝐴𝐷|Lwµg𝐴𝐵𝐶𝐷ˆćg
      
      
D~Ö𝐴𝐵= 𝐷𝐶LÔ𝐴𝐶⊥𝐵𝐷Lwµg𝐴𝐵𝐶𝐷ˆĈg
 2   !'()*"+,-.
» 2V2024·ĉÿĊċ··íWqµg ABCD tLELF Bæˆ ADLBC "t<LwĂă !Œ𝐴𝐵+
𝐷𝐶”GH"XV    W
A~2𝐸𝐹
B~𝐴𝐶+ 𝐷𝐵
C~𝐸𝐵+ 𝐸𝐶
D~𝐹𝐴+ 𝐹𝐷
õh 2-1V2024·QR·íîïðWHµ·¸g𝐴𝐵𝐶"Ččˆ𝑂L<𝐷X/Ü𝐵𝐶Ďď]𝐵"·HB<Lw
𝐴𝐷= V    W
A~𝑂𝐵+
2
3𝐴𝐶
B~
1
2𝑂𝐵+
2
3𝐴𝐶
C~𝑂𝐵+
3
4𝐴𝐶
D~
1
2𝑂𝐵+
3
2𝐴𝐶
õh 2-2V2023·ĐđĒÿ·¾íWq △𝐴𝐵𝐶tL𝐴𝐵= 4,𝐴𝐶= 6L< DLE Bæq/Ü𝐴𝐵L𝐴𝐶ĎLÔ D
ˆ𝐴𝐵t<L𝐴𝐸=
1
2𝐸𝐶LÖ𝐴𝑃= 𝐴𝐷+ 𝐴𝐸Lwē/𝐴𝑃ĔÆ △𝐴𝐵𝐶"V    W
A~Zč
B~ĕč
C~ãč
D~Čč
õh 2-3V2024·Ėė·íîïðWHĘ △𝐴𝐵𝐶tL∠𝐵= ∠𝐶= 30°,𝐴𝐵= 1LD ˆ/Ü𝐴𝐵Ď"ę<LÆ D
Ě𝐷𝐸∥𝐵𝐶ě𝐴𝐶§ E~Æ D Ě𝐷𝐹⊥𝐵𝐶ě𝐵𝐶§ FLw|2𝐵𝐹+ 𝐷𝐸| = V    W
A~ 3
B~2 3
C~3 3
D~5 3
 3   !"/012
» 3V2023·ĜÿĝĞ·íîïðWĂă !›htLĈğ"XV    W
A~𝑀𝑁= 𝑁𝑀
B~𝐴𝐵+ 𝐴𝐶= 𝐵𝐶
C~𝐴𝐵+ 𝐶𝐴= 𝐵𝐶
D~𝑀𝑁+ 𝑁𝑃+ 𝑃𝑄= 𝑀𝑄
õh 3-1V2023·ĜÿĝĞ·íîïðW½x !𝑎,𝑏Lw2 𝑎+ 𝑏−𝑎−𝑏= V    W
A~𝑎+ 𝑏
B~𝑎−𝑏
C~3𝑎+ 𝑏
D~𝑎+3𝑏
õh 3-2V2024·QR·íîïðWq △𝐴𝐵𝐶tL𝑁𝐴+ 𝑁𝐶= 0,𝐵𝑀= 2𝑀𝐶LwV    W
A~𝑁𝑀= −
1
3𝐴𝐵−
1
6𝐴𝐶
B~𝑁𝑀=
1
3𝐴𝐵−
1
6𝐴𝐶
C~𝑁𝑀= −
1
3𝐴𝐵+
1
6𝐴𝐶
D~𝑁𝑀=
1
3𝐴𝐵+
1
6𝐴𝐶
õh 3-3V2024·ĠġĢ·¾íWģ—ĤF" △𝐴𝐵𝐶tL<𝐷X/Ü𝐵𝐶Ďď]𝐵"·HB<L<𝐸X/Ü
𝐴𝐵"t<Lw𝐷𝐸= V    W
      
      
A~−
1
3𝐴𝐵−
1
6𝐴𝐶
B~−
1
6𝐴𝐵−
1
3𝐴𝐶
C~−
5
6𝐴𝐵−
1
3𝐴𝐶
D~−
5
6𝐴𝐵+
1
3𝐴𝐶
 4  34 !/01256

» 4V2023·ĥĦħĨĩ·äíWģ—L½x △𝐴𝐵𝐶tL𝐷X𝐴𝐵µĎ¾<LÖ𝐷𝐵=
1
2𝐴𝐷L3𝐶𝐷= 𝐶𝐴+𝑚
𝐶𝐵Lw𝑚= V    W
  
A~−2
B~2
C~−1
D~3
õh 4-1V2023·Īī·íîïðW½xq △𝐴𝐵𝐶tL< D ˆµ BC "t<LÖ𝐴𝐷+ 𝐵𝐶= 𝜆𝐴𝐵+𝜇𝐴𝐶Lw
𝜆−𝜇= V    W
A~1
B~Ĭ1
C~2
D~Ĭ2
õh 4-2V2024·ĩĭĮt·íîïðWģ—Lqzµg𝐴𝐵𝐶𝐷tL𝑀ˆ𝐵𝐶"ď]<𝐶"·HB<L𝐴𝐶
Œ𝑀𝐷Gě§<𝑃LÖ𝐴𝑃= 𝑥𝐴𝐵+𝑦𝐴𝐷Lw𝑥𝑦= V    W
A~
2
3
B~
9
16
C~
3
4
D~
4
9
õh 4-3V2023·įøİı·íîïðWq △𝐴𝐵𝐶tL𝐷X/Ü𝐵𝐶Ď¾<Léê𝐵𝐷= 2𝐷𝐶,𝑀X/Ü𝐴𝐷"
t<Lü𝐵𝑀= 𝑥𝐴𝐵+𝑦𝐴𝐶LwV    W
A~𝑥−𝑦= −
1
2
B~𝑥+ 𝑦= −
1
2
C~𝑥−𝑦=
1
2
D~𝑥+ 𝑦=
1
2
 5   !7/89:;
» 5V2024·QR·íîïðW½x !𝑎L𝑏Lw“𝑎//𝑏”X“IJq𝜆∈𝑅LijË𝑎= 𝜆𝑏”"V    W
      
      
A~ó=”òBëì
B~òB”ó=ëì
C~ò=ëì
D~ô”òB†”ó=ëì
õh 5-1V2024·ĎĴĵā·¾íWü𝑂ˆ △𝐴𝐵𝐶ĤqĎ¾<.Ö×
 x(y(z éê𝑥𝑂𝐴+𝑦𝑂𝐵+𝑧𝑂𝐶=
0 𝑥2 + 𝑦2 + 𝑧2 ≠0 Lw“𝑥𝑦𝑧= 0”X“<𝑂q △𝐴𝐵𝐶"µĤqē/Ď”"V    
W
A~òB”ó=ëì
B~ó=”òBëì
C~òBó=ëì
D~ô”òBĶ”ó=ëì.
õh 5-2V2023·ö÷Ĵķ·äíW½x𝑎,𝑏XZπ„… !LĸĹ“𝑎∥𝑏”X“IJq𝜆≠0LijË|𝑎+𝜆𝑏
| = |𝑎| + |𝜆𝑏|”"V   W
A~òBĺ”ó=ëì
B~ó=ĺ”òBëì
C~òBó=ëì
D~ô”òB†”ó=ëì
õh 5-3V2023·ĻļĽľ·¾íW½xĈ·¸g𝐴𝐵𝐶"µĿˆ 6L 𝐴𝑃= 𝜆𝐴𝐵+𝜇𝐴𝐶L𝜆∈[0,1]L𝜇∈[0,1]
Ô3𝜆+ 4𝜇= 2Lw<𝑃´ē/𝐵𝐶ŀŁ"łà̈V    W
A~2 3
B~3
C~3 3
D~3 3
2
¾(ž
1~V2023·ö÷àı··íWü𝑎L𝑏X„… !L“
𝑎
|𝑎| =
𝑏
|𝑏|”X“𝑎= 𝑏”"V    W
A~òBĺ”ó=ëì
B~ó=ĺ”òBëì
C~òBó=ëì
D~ô”òB†”ó=ëì
2~V2023·ŃÇÿ·íîïðW½xĈg ABCD "µĿˆ 1L< M éê𝐴𝐵+ 𝐵𝐶= 2𝐴𝑀Lw|𝑀𝐷| =
V    W
A~
1
2
B~1
C~ 2
2
D~ 2
3~V2023·Ġÿò·¾íW½xĈg𝐴𝐵𝐶𝐷"µĿˆ 1Lw|𝐴𝐵+ 𝐵𝐶−𝐶𝐴| = V    W
A~0
B~ 2
C~2 2
D~4
4~V2024·øùÿÅ·íîïðWqýg𝐴𝐵𝐶𝐷tL𝐴𝐵//𝐶𝐷LÔ𝐴𝐵= 2𝐶𝐷L<𝑀X𝐵𝐶"t<Lw𝐴𝑀=
V    W
A~
2
3𝐴𝐵−
1
2𝐴𝐷
B~
1
2𝐴𝐵+
2
3𝐴𝐷
C~𝐴𝐵+
1
2𝐴𝐷
D~
3
4𝐴𝐵+
1
2𝐴𝐷
      
      
5~V2024·Ėĭ·íîïðWq △𝐴𝐵𝐶tL𝐴𝐵= 4𝐴𝐷L𝐶𝐸= 2𝐸𝐷.Ö𝐵𝐶= 𝜆𝐴𝐸+𝜇𝐶𝐷LwV    W
A~𝜆+ 𝜇= 5
B~𝜆−𝜇= 1
C~𝜆𝜇= 6
D~
𝜆
𝜇= 3
6~V2024·ŃÇŃī·íîïðW½x⃗𝑒1,⃗𝑒2Xπ”7/" !LÖ2
→𝑒1 +𝜆
→𝑒2Œ𝜇
→𝑒1 +
→𝑒2X7/ !Lw
V    W
A~
𝜆
𝜇= −2
B~𝜆𝜇= −2
C~
𝜆
𝜇= 2
D~𝜆𝜇= 2
7~V2024·įø·íîïðW½x !𝑒1L𝑒2XĎπ”7/"žŸ !LÔ𝐴𝐵= 𝑒1 +2𝑒2L𝐵𝐶= −3𝑒1
+2𝑒2L𝐷𝐴= 3𝑒1−6𝑒2LwV    W
A~𝐴(𝐵(𝐶·<7/
B~𝐴(𝐵(𝐷·<7/
C~𝐴(𝐶(𝐷·<7/
D~𝐵(𝐶(𝐷·<7/
8~V2024·QR·äíW<𝑂,𝑃X △𝐴𝐵𝐶ĤqZπ”œ"<Léê𝑂𝑃= 𝑂𝐴+ 𝑂𝐵+ 𝑂𝐶Lwē/𝑂𝑃ĔÆ
△𝐴𝐵𝐶"V    W
A~ãč
B~ĕč
C~Zč
D~Čč
ä(ń
9~V23-24 ¾Ă·ŅņŇňù·ʼntWĂăŊ*tĈğ"XV    W
A~Ö𝑎Œ𝑏ŋXžŸ !Lw𝑎= 𝑏
B~… !"Ŀkˆ…L XÝ-"
C~Ö𝑎Œ𝑏Xz !Lw𝑎= 𝑏
D~Ö𝑎+ 𝑏= 0¶𝑎−𝑏= 0Lw𝑎//𝑏
10~V2024·Ōĥ·äíW △𝐴𝐵𝐶"ãčˆ<𝐺L< OLP X △𝐴𝐵𝐶ĤqZπ”œ"<Léê𝑂𝑃= 𝑂𝐴+
𝑂𝐵+ 𝑂𝐶LwV    W
A~𝑂,𝑃,𝐺·<7/
B~𝑂𝑃= 2𝑂𝐺
C~2𝑂𝑃= 𝐴𝑃+ 𝐵𝑃+ 𝐶𝑃
D~<𝑃q △𝐴𝐵𝐶"Zō
11~V2023·ĴÿŎēŏŐőžŸ·íîïðW
ŒpŒIJqœŔŕ›LŖńpŒt"íńŗ§
"Ř
ř~½xŚÇś"ŜŝŞşŠĈšµgˆbŢLţcĈšµg"ŤëµĚˆĈg"¾ëµÀÁ¬š
€ĈgLģ—ĤFLwqť—gtLĂăŊ*Ĉğ"XV    W
      
      
  
A~𝐺𝐻=
2 3
3 + 1 𝐵𝐷
B~𝐵𝐸= 𝐵𝐷+
3
2 𝐶𝐹
C~𝐺𝐵=
3
3 𝐵𝐷−
1
2𝐶𝐹
D~𝐼𝐶= 3+ 3
6 𝐵𝐷+
3−1
4 𝐶𝐹
·(ef
12~V2023·Ŧŧø·íîïðWqzµg𝐴𝐵𝐶𝐷tL3
→
𝐵𝐸=
→
𝐸𝐷L
→
𝐶𝐸= 𝜆
→
𝐴𝐵+𝜇
→
𝐴𝐷(𝜆,𝜇∈R)L2𝜆+ 𝜇=      
.
13~V23-24 ¾Ă·ĎĴŨė·ʼntWĂ㧠!"ũLŪ	Ĉğ"X      
.
①… !z§Ý- !²
②v§„… !𝑎,𝑏LÖ𝑎//𝑏Lw𝑎=±  𝑏²
③v§„… !𝑎,𝑏LÖ𝑎=±  𝑏Lw𝑎//𝑏²
④v§„… !𝑎,𝑏LÖ𝑎//𝑏Lw𝑎Œ𝑏Ĥqē/¾8ãū.
14~V2024·ĩĭŬ’··íWŭŮXůRŰű
Ų(ųŲLàŴqŵ 222 PLŭوŶŷŸ2ĔŹ
¾źĚŪ“LŻİN “żŽž—”Lſƀ“ŭŮƁ—” (cē¸·¸g"ƂµˆµË´"Ĉg). ƃn “ŭŮ
Ɓ—”LÀÁģ—ĤF"—gL‰XƄ·€QH"·¸gŒtƅ"¾€áHµ·¸gƆº"¾€àHµ·¸
gLÔ𝐷𝐹= 𝐴𝐹L<𝑃q𝐴𝐵ĎL𝐵𝑃= 2𝐴𝑃L<𝑄q △𝐷𝐸𝐹 Z (IµƇ)¾<LÖ𝑃𝑄= 𝜆𝑃𝐷+ 𝑃𝐴Lw𝜆"łà
̈      
. 
(Dƈ
15~V23-24 ¾Ă·ŅƉƊ·ʼntWƋĂăƌh
(1)(𝐴𝐵+ 𝑀𝐵) + (−𝑂𝐵−𝑀𝑂)²
      
      
(2)𝐴𝐵−𝐴𝐷−𝐷𝐶²
(3)𝑂𝐴−𝑂𝐷+ 𝐴𝐷²
16~V24-25 ä·ĎĴ·ƍʼnĚƎWģ—LE(F(G ÐƏXĈ·¸g ABC "µ AB(BC(AC "t<.
(1)qc A(B(C(E(F(G ˆ<¶Ɛ<" !tL«¬Œ !𝐸𝐹7/" !²
(2)qc A(B(C ˆ<Lc E(F(G ˆƐ<" !tL«¬Œ !𝐺𝐹íGH" !²
(3)qc E(F(G ˆ<Lc A(B(C ˆƐ<" !tL«¬Œ !𝐸𝐺GH" !.
17~V23-24 ¾Ă·ĉÿŷƑ·ƒÜ©sWģ—L<𝐷X △𝐴𝐵𝐶t BC µ"t<L𝐴𝐵= 𝑎,𝐴𝐶= 𝑏.
(1)Ö<𝑂X △𝐴𝐵𝐶"ãčLƓ𝑎,𝑏EF𝐴𝑂;
(2)Ö<𝑂X △𝐴𝐵𝐶"ãčL5𝑂𝐴+ 𝑂𝐵+ 𝑂𝐶.
      
      
18~V23-24 ¾Ď·įøƔī·ʼnƕWü𝑎,𝑏X”7/"π„… !~
(1)Ö𝑂𝐴= 4𝑎−2𝑏,𝑂𝐵= 6𝑎+2𝑏,𝑂𝐶= 2𝑎−6𝑏L5Ɩ
𝐴,𝐵,𝐶·<7/²
(2)Ö4𝑎+
1
2𝑘𝑏Œ
1
2𝑘𝑎+ 𝑏7/L5×
 k "Ì~
19~V23-24 ¾Ď·ŌĥàƗ·ʼnƕWģ—ĤFL½x<𝐺X △𝐴𝐵𝐶"ãčLÆ<𝐺Ěē/B挵𝐴𝐵(𝐴𝐶
ě§𝑀(𝑁Ï<V<𝑀(𝑁Œ<𝐵(𝐶”ãūWLü𝐴𝐵= 𝑥𝐴𝑀L𝐴𝐶= 𝑦𝐴𝑁~
(1)5𝑥+ 𝑦"̲
(2)5
1
𝑥−1 +
2
𝑦−1"łáÌLM5Ƙ“𝑥L𝑦"Ì~