平面向量题型训练 【重点知识点同步练习】 1. 下列命题: (1) 若 ab\uf03d\uf072\uf072, 则 ab\uf03d\uf072\uf072。(2) 两个向量相等的充要条件是它们的起点相同, 终点相同。(3) 若 AB\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072。(5) 若aDC\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072, 则\uf072ABCD 是平行四边形。(4) 若 ABCD 是平行四边形, 则 AB其中正确的是_______ (1,1),ab\uf03d\uf03d(1, 1),c\uf02d(2) 下列向量组中, 能作为平面内所有向量基底的是 \uf075\uf072\uf075\uf075 \uf072 B. 1\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072分别是 ABC\uf044DC\uf03d,b bc\uf03d\uf03d\uf072\uf072 \uf072\uf072, 则 ac\uf03d\uf072\uf072。(6) 若 // , //a b b c\uf072\uf072 \uf072 \uf072, 则//ac\uf072。2. (1) 若\uf072\uf072( 1,2)\uf03d \uf02d\uf072, 则 c \uf03d\uf072______ A. 12(0,0),(1, 2)\uf02dee\uf03d\uf03d2( 1,2),\uf03d \uf02d(5,7)ee\uf03d\uf075\uf072\uf075\uf075 \uf072 C. 1\uf075\uf075\uf075\uf0722(3,5),(6,10)ee\uf03d\uf072 \uf075\uf075\uf075 \uf072\uf03d\uf075\uf072\uf075\uf075 \uf072 D. 1213(2, 3),\uf02d( ,2)4ee\uf03d\uf03d\uf02d\uf075\uf072\uf075\uf075 \uf072 (3) 已知,AD BE的边,BC AC 上的中线, 且,ADa BEb\uf03d\uf0beAB\uf03d\uf0be\uf0ae\uf0beAC\uf072, 则 BC\uf075\uf075\uf075 \uf072可用向量 , a b\uf072 \uf072表示为_____ (4)ABC\uf044中, 点 D 在 BC 边上, 且\uf0be\uf0ae\uf0beAB1(1, ),(0,2\uf072\uf072 \uf072\ue000\uf0be\uf0ae\uf0beDB\uf0be\uf0ae\uf0beCD\uf03d2\uf0be\uf0ae\uf0beBC\uf072,\uf0be\uf0ae\uf0be\uf0ae\uf0beCD\uf02b\uf03dsr, 则sr \uf02b 的值是___ 3. (1) △ABC 中,3||\uf03d,4\uf072 \uf075 \uf072||\uf072\uf03d\uf0be\uf0ae\uf0beAC,5||\uf03d, 则\uf072与 d\uf03d\uf0d7BCAB\uf075 \uf072_________ \uf070, 则 k 等于____ (2) 已知1),,2abcakb dab\uf03d\uf03d\uf02d\uf03d\uf02b\uf03d\uf02d\uf072\uf072\uf072\uf072, c的夹角为4(3) 已知2,5,3a\uf072 \uf072ba b\uf03d\uf03d\uf03d \uf02d\uf072, 则 ab\uf072\uf02b\uf072\uf072等于____ \uf072(4) 已知 , a b是两个非零向量, 且 abab\uf03d\uf03d\uf02d\uf072\uf072, 则 与aab\uf02b\uf072\uf072\uf072的夹角为____ 4. 已知3\uf0aea||\uf03d\uf0aea,5||\uf03d\uf0aeb, 且12\uf03d\uf0d7\uf0aeb\uf0aea, 则向量\uf0aea 在向量\uf0aeb 的夹角为锐角, 则\uf06c 的取值范围是______ 32\uf072之间有关系式kab\uf02b\uf03d\uf0aeb 上的投影为______ 5. (1) 已知)2 ,(\uf06c\uf06c\uf03d,) 2 ,3 ( \uf06c\uf0be\uf0ae\uf0beOF\uf03d\uf0aeb, 如果\uf0aea 与(2)OFQ\uf044的面积为 S , 且1\uf03d\uf0d7\uf0be\uf0ae\uf0beFQ, 若21\uf03c\uf03c S, 则\uf0be\uf0ae\uf0beFQ\uf0be\uf0ae\uf0beOF,夹角\uf071 的取值范围是_____ \uf072\uf072其中(3) 已知(cos ,sin ),x(cos ,sin ),yax b\uf072与 by\uf03d\uf03d\uf072\uf072\uf072a\uf072与 b3,0akbk\uf02d\uf03e\uf072\uf072, ①用 k 表示a b \uf0d7\uf072 \uf072; ②求 a b \uf0d7\uf072 \uf072的最小值, 并求此时a的夹角\uf071 的大小 6. (1) 化简: ① AB(2) 若正方形 ABCD 的边长为 1,BCCD\uf02b\uf02b\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072___; ② AB\uf075\uf075\uf075 \uf072ADDC\uf072 \uf075\uf075\uf075\uf072\uf02da BC\uf075\uf075\uf075 \uf072\uf02d\uf03d\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072____; b AC\uf03d\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072,,ABc\uf03d\uf072 \uf075\uf075\uf075 \uf072\uf072, 则|\uf075\uf075\uf075\uf072|ab\uf075\uf075\uf075 \uf072c\uf02b \uf02b\uf072\uf072\uf072=_____ (3) O 是三角形 ABC 所在平面内一点, 满足2OBOCOBOCOA\uf02d\uf03d\uf02b\uf02d, 则三角形 ABC 的形状为____ (4) 若 D 为 ABC\uf044的边 BC 的中点,ABC\uf044所在平面内有一点 P , 满足0PABP CP\uf02b\uf02b\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf072, 设||||AP\uf075\uf075\uf075 \uf072PD\uf06c\uf03d\uf075\uf075\uf075 \uf072, 则\uf06c 的值为___ (5) 若点 O 是ABC△A的外心, 且(2,3), (5,4)B0\uf075\uf075\uf075 \uf072OAOBCO\uf075\uf075\uf075 \uf072\uf02b\uf02b\uf03d\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf072, 则ABABC\uf075\uf075\uf075\uf072△的内角 C 为____ )R\uf0ce, 则当 \uf06c =____时, 点 P 在第一、 三象限的角平分线上 7. (1) 已知点,\uf075\uf075\uf075 \uf072(7,10)C, 若(\uf06cAPAC\uf06c\uf02b(2) 已知1(2,3), (1,4),B(sin ,cos )x2AABy\uf03d且,,(,)2 2x y\uf070 \uf070\uf0ce \uf02d, 则 xy\uf02b\uf03d 8. 设 (2,3), ( 1,5)AB \uf02d, 且13ACAB\uf03d\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072,3ADAB\uf03d\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072, 则 C、 D 的坐标分别是__________ 9. 已知向量a =(sinx, cosx) , b =(sinx, sinx) , c =(-1, 0)。 \uf070, 求向量a 、 c 的夹角; (2) 若 x∈\uf072 \uf072均为单位向量, 它们的夹角为 60\uf06f, 那么|\uf0ae\uf0ae\uf0ae\uf0ae\uf0ae\uf02d\uf0d7\uf03d\uf02d\uf0d7bacba)(\uf0ae\uf0ae\uf0ae\uf02d\uf0d7\uf02b; ④ 若0\uf03d\uf0d7ba, 则(1) 若 x=3]43 |b,8a3\uf070[\uf070\uf02d, 函数baxf\uf0d7\uf03d \uf06c)(的最大值为21, 求\uf06c 的值。 10. 已知 , a b\uf02b\uf0aec\uf075\uf075 \uf072\uf072=_____ \uf0ae\uf0ae\uf0d7ba)\uf072 \uf07211. 下列命题中: ① \uf0aec\uf0aea\uf0d7; ② \uf0aec\uf0aeb\uf0aea\uf0d7\uf03d\uf0d7\uf0d7()(; ③ 2()a b\uf02d\uf072; ⑥\uf0ae\uf0ae2|| a\uf0ae\uf03d 22|| ||||abb\uf0ae\uf0ae0\uf03d\uf0aea或0\uf03d\uf0aeb; ⑤若,a b\uf0d7c b\uf03d \uf0d7\uf072 \uf072则 ac\uf03d\uf07222aa\uf03d\uf072\uf072; ⑦2a b\uf0d7\uf072b\uf072aa\uf03d\uf072 \uf072\uf072; ⑧222()a b\uf0d7ab\uf03d\uf0d7\uf072 \uf072\uf072\uf072;⑨222()2abaa b\uf0d7 \uf02bb\uf072\uf02d\uf03d\uf02d\uf072\uf072\uf072\uf072 \uf072\uf072。 其中正确的是_____ 12. (1) 若向量( ,1),x\uf072(4, )abx\uf072\uf03d\uf03dx\uf072, 当 x=_____时a\uf072\uf072,\uf072与b\uf072共线且方向相同 \uf072\uf072, 则 x=______ (2) 已知(1,1),(4, )ab\uf03d\uf03d\uf072,2uab\uf03d\uf02b2vab\uf03d\uf02b\uf072\uf072\uf072, 且 //uv (3) 设( ,12),k\uf075\uf075\uf075 \uf072(4,5),\uf075\uf075\uf075 \uf072(10, )PAPBPCk\uf075\uf075\uf075 \uf072\uf03d\uf03d\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072, 则 k=_____时, A, B, C 共线 \uf075\uf075\uf075 \uf072, 则 m \uf03d 13. (1) 已知(2) 以原点 O 和 A(4, 2) 为两个顶点作等腰直角三角形 OAB,( , ),na b\uf03d向量nm\uf05e( 1,2),\uf03d \uf02d(3, )OAOBm\uf03d, 若OAOB\uf05e90B\uf0d0\uf03d\uf0b0 , 则点 B 的坐标是_____ (3) 已知\uf072\uf072\uf075\uf072, 且 nm\uf03d\uf072\uf075\uf072, 则 m\uf075\uf072的坐标是________ 14. 若⊿ABC 三边中点分别为(2, 1)、(-3, 4)、(-1, -1), 则⊿ABC 的重心的坐标为 【平面向量高考经典试题】 一、 选择题 ( 5,6)a \uf03d \uf02d,(6,5)b \uf03d, 则 a与bA. 垂直 B. 不垂直也不平行 C. 平行且同向 D. 平行且反向 2、 已知向量(1)( 1)nn\uf03d\uf03d \uf02d,,,ab, 若2 \uf02da1. 已知向量\uf072\uf072\uf072\uf072 b 与b 垂直, 则\uf03da( ) A. 1 B.2 \uf072 \uf072 \uf072的夹角为 60°, 则 a a C. 2 D. 4 \uf072 \uf0723、 若向量 , a b\uf072 \uf072满足|| |\uf03d| 1\uf03dab\uf072,, a ba b\uf0d7 \uf02b \uf0d7\uf072 \uf072=______; 答案:32; \uf0724、 设两个向量22(\uf06c2,cos)a\uf06c\uf061\uf03d\uf02b\uf02d\uf072和( ,msin ),\uf0612mb\uf03d\uf02b\uf072其中 , , m\uf06c\uf061 为实数. 若2 ,ba\uf03d\uf072则m\uf06c的取值范围是( ) A. [ 6,1]\uf02d B. [4,8] C. (,1]\uf02d\uf0a5 D. [ 1,6]\uf02d22\uf03d\uf02d 【答案】 A 由22(\uf06c2,cos)a\uf06c\uf061\uf03d\uf02b\uf02d\uf072,( ,msin ),\uf0612mb\uf03d\uf02b\uf0722 ,ba\uf03d\uf072\uf072可得22cos2sinmm\uf06c\uf06c\uf0ee\uf061\uf061\uf02b\uf0ec\uf0ed\uf03d\uf02b, 设km\uf06c\uf03d代入方程组可得22222cos2sinkmmk mm\uf061\uf061\uf02b\uf03d\uf0ec\uf0ed\uf0ee\uf0e6\uf0e7\uf0e8\uf02d\uf03d\uf02b消去 m 化简得2222cos2sin22kkk\uf061\uf061\uf0e6\uf0e7\uf0e8\uf0f6\uf0f7\uf0f8\uf02d\uf03d\uf02b\uf02d\uf02d, 再化简得22422cos2sin022kk\uf061\uf061\uf0f6\uf0f7\uf0f8\uf02b\uf02d\uf02b\uf02d\uf03d\uf02d\uf02d再令1\uf02d2tk\uf03d代入上式得222(sin1)(16182)0tt\uf061 \uf02d\uf02b\uf02b\uf02b\uf03d可得2(16182) [0,4]\uf0cett\uf02d\uf02b\uf02b解不等式得18[ 1,]t\uf0ce \uf02d \uf02d因而1\uf02d1812k\uf02d \uf0a3\uf0a3 \uf02d解得 61k\uf02d \uf0a3\uf0a3 . 故选 A 5、 在直角 ABC\uf075\uf075\uf075\uf072\uf0442中, CD 是斜边 AB 上的高, 则下列等式不成立的是 \uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072 (B) BCBA BC\uf03d\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072(A)ACAC AB\uf0d7\uf03d2\uf0d7\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075 \uf072 (C)2ABAC CD\uf0d7\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072 (D) 22() (\uf0b4\uf075\uf075\uf075 \uf072)AC AB\uf0d7BA BC\uf0d7CDAB\uf03d\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075 \uf072 6、 在∆ABC 中, 已知 D 是 AB 边上一点, 若 AD =2DB , CD =CB\uf06cCA\uf02b32 1,则\uf06c= (A)32 (B) 31 (C) -31 (D) -3( )f x2ex\uf02b 7 把函数exy \uf03d2 的图像按向量(2)\uf03d, 0a平移, 得到2ex\uf02d y\uf03d的图像, 则( )f x \uf03d ( ) A. ex\uf02bB. eABC2x\uf02d C.D.8、 已知O是\uf075\uf075\uf075\uf072△所在平面内一点, D 为 BC 边中点, 且2OA OB OC2AOOD\uf03d D . 2AOOD\uf03d 所在平面内一点, D 为 BC 边中点, ∴ 9、 对于向量, a 、 b、 c和实数 , 下列命题中真命题是 A 若, 则 a=0 或 b=0 B 若C 若=, 则 a=b 或 a=-b D 若10、 设 , ab 是非零向量, 若( )(f xx\uf03dA.⊥ab B.∥ab C. |xy\uf0e6\uf0f6\uf03d\uf02b\uf0e7\uf0f7\uf0e8\uf0f8π2cos234\uf0e7\uf0f7\uf0e8\uf0f8\uf02b\uf02b\uf03d 0\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072, 则( ) A. AOOD\uf075\uf075\uf075\uf072\uf03d\uf03d\uf075\uf075\uf075\uf072 B .\uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072 C .3AOOD\uf075\uf075\uf075\uf072 \uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072解析: O是ABC△2OBOCOD\uf02b\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072 , 则 λ=0 或 a=0 , 则 b=c )x\uf02dab 的图象是一条直线, 则必有( ) |b D. || ||\uf0b9ab \uf0f6\uf02d\uf0f7\uf0f8π234\uf0f7\uf0f8) (\ue000| |\uf03dπ,\uf02baba11、 将π2cos36的图象按向量24\uf0e6\uf0e7\uf0e8\uf03d \uf02da平移, 则平移后所得图象的解析式为( ) A.xy\uf0e6\uf0f6\uf03d\uf02b\uf02d B .2cosxy\uf0e6\uf0e7\uf0e8\uf0f6\uf03d\uf02d\uf02b B A \uf03d \uf02dC D C .π2cos2312xy\uf0e6\uf0e7\uf0e8\uf0f6\uf0f7\uf0f8\uf03d\uf02d\uf02d D .π2cos2312xy\uf0e6\uf0e7\uf0e8\uf0f6\uf0f7\uf0f8\uf03d\uf02b\uf02b 12、 设 a=(4,3),a 在 b 上的投影为225,b 在 x 轴上的投影为 2,且|b|<1,则 b 为 A.(2,14) B.(2,- 7b 满足2) C.(-2, 72) D.(2,8) 13、 若非零向量 , a\uf02b\uf03dabb , 则( ) A.2\uf03e \uf032 \uf02baab B .22\uf03c\uf02baab C .2\uf03e\uf02b\uf032bab D . 22\uf03c\uf02bbab 【答案】: C 2,\uf02b\uf032\uf03d\uf02b\uf0a3\uf02b\uf03d\uf051 abab+ ba + bbb 由于 , ab 是非零向量, 则必有\uf0b9a + bb, 故上式中等号不成立 。 ∴22\uf03e\uf02bbab 14、 已知平面向量(1 1), ,(11)\uf03d\uf03d\uf02d,ab, 则向量1322\uf02d\uf03dab( ) A. ( 21)\uf02d\uf02d, \uf075\uf075\uf075 \uf072B . ( 21)\uf02d , C . ( 1 0)\uf075\uf075\uf075\uf072\uf02d ,OC \uf075\uf075\uf075 \uf072 \uf075\uf075\uf075\uf072D . ( 1 2)\uf075\uf075\uf075 \uf072\uf02d ,, 15、 已知向量(4,6),(3,5),OAOB\uf03d\uf03d2\uf075\uf075\uf075 \uf072且,//OA ACOB\uf05e则向量OC2\uf075\uf075\uf075\uf072等于 (A)\uf0f7\uf0f8\uf0f6\uf0e7\uf0e8\uf0e6\uf02d72,73 (B)\uf0f7\uf0f8\uf0f6\uf0e7\uf0e8\uf0e6\uf02d214,7 (C)\uf0f7\uf0f8\uf0f6\uf0e7\uf0e8\uf0e6\uf0e7\uf0e8\uf0e6\uf02d72,73 (D)\uf0f7\uf0f8\uf0f6\uf0e7\uf0e8\uf0e6\uf02d214,7 16、 若向量a 与b 不共线,0\uf0b9\ue000a b, 且\uf0f6\uf0f7\uf0f8\ue000\ue000a ac = a -ba b, 则向量a 与c 的夹角为( ) A. 0 B.π6 C.π3 D.π2 17. 若A. ( 1( )f x2) y\uf03d的图按向量a 平移后得到函数B. (12)\uf02d, (2, )Bb ,(4,5)C(1)2yf x\uf03d\uf02bD. (1 2)\uf02d的图象, 则向量a = ( ) \uf075\uf075\uf075 \uf072与OB\uf02d \uf02d,C. ( 1 2)\uf02d , ,18、 设 ( ,1)足的关系式为( ) (A) 45a\uf02d解析: 选 A. 由 OA二、 填空题 A a,为坐标平面上三点, O为坐标原点, 若OA\uf075\uf075\uf075 \uf072在OC\uf075\uf075\uf075\uf072方向上的投影相同, 则a 与b 满3\uf075\uf075\uf075 \uf072b\uf03d (B) 5与OB43ab\uf02d\uf03d (C) 4方向上的投影相同, 可得: OA OC514\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075\uf072ab\uf02b\uf03d (D) 5\uf0d7\uf03d414ab\uf02b\uf03d \uf075\uf075\uf075 \uf072在OC\uf075\uf075\uf075\uf072OB OC\uf0d7\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075\uf072 1、 如图, 在 ABC\uf044中,120 ,2,1,BACABACD\uf0d0\uf03d\uf0b0\uf03d\uf03d是边 BC 上一点,2,DCBD\uf03d则 AD BC \uf03d\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072\ue000__________ .【答案】83\uf02d 【分析】 法一: 由余弦定理得222222cos222ABACBCABADBDBABBDAC\uf02b\uf0b4ABBD\uf02b\uf0b4\uf02d\uf02b\uf0b4\uf02d\uf03d\uf03d\uf0b42\uf0b4可得7BC \uf03d133,AD \uf03d, 又,AD BC\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072夹角大小为ADB\uf0d0,23298cos294 13791ADABADBBDAD\uf02d\uf0d0\uf03d\uf03d \uf02d\uf0b4\uf0b4\uf0b4, 所以 AD BC \uf03d\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072\ue0008cos3ADBCADB\uf0b4\uf0b4\uf0d0\uf03d \uf02d. 法二: 根据向量的加减法法则有: BC\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072AC1AB\uf03d\uf02d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075 \uf07212()33133ADABBDABACABACAB\uf03d\uf02b\uf03d\uf02b\uf02d\uf03d\uf02b\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072, 此时 2213232323()()AD BC·ACAB ACABACAC AB·AB\uf03d\uf02b\uf02d\uf03d\uf02b\uf02d\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075\uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072 18183333\uf03d\uf02d\uf02d\uf03d \uf02d. 2、 在四面体 O-ABC 中,3、 已知向量\uf072 \uf072, 的夹角为,,,OAa OB11\uf028\uf029,\uf072b OC(\uf05ec D)\uf03d\uf03db\uf03d\uf06c\uf072\uf075\uf075\uf075 \uf072\uf072 \uf075\uf075\uf075 \uf072\uf072 \uf075\uf075\uf075\uf072\uf072为 BC 的中点, E 为 AD 的中点, 则OE = (用 a, b, c 表示) a + b , 则实数\uf06c 的值是 【答案】 -3. \uf072\uf072\ue000 . 【答案】22 4, ,\uf028\uf029a =b =. 若4、 若向量a b\uf06f60 ,1ab\uf03d\uf03d\uf072, 则 \uf028\uf029aab\uf02d\uf03d1 5、 如图, 在AC\uf03dABC△, 则 m中, 点O是 BC 的中点, 过点O的直线分别交直线 AB , AC 于不同的两点 Mn\uf02b的值为 . N,, 若 ABmAM\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf075 \uf072,nAN\uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072ABDC 解析: 由 MN 的任意性可用特殊位置法: 当 MN 与 BC 重合时知 m=1, n=1, 故 m+n=2 6、 在平面直角坐标系中, 正方形 OABC 的对角线OB 的两端点分别为 三、 解答题: 1、 已知△ ABC顶点的直角坐标分别为5\uf03dc, 求 sin∠ A 的值; (2) 若∠ A 是钝角, 求c 的取值范围. 2、 已知ΔABC三个顶点的直角坐标分别为A(3, 4) 、 B(0, 0) 、 C(c , 0) . (1) 若0AB AC \uf03d\ue000, 求c 的值; (2) 若5c \uf03d, 求 sin∠A 的值 (0 0)O ,,(11)B ,, 则 AB AC \uf03d\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075\uf072\ue000 ) 0 ,c() 0 , 0 (B) 4 , 3 (AC、、. (1) 若3、 在ABC△中, 角 A B C,\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075 \uf072\ue000,的对边分别为5, 且tan3 7abcC \uf03d,,,. (1) 求cosC (2) 若2CB CA \uf03d9ab\uf02b \uf03d, 求c . 【体验练习题一: 】 一、 选择题 1. 已知平面向量 a=A 平行于 x轴 B. 平行于第一、 三象限的角平分线 C. 平行于 y 轴 D. 平行于第二、 四象限的角平分线 ,1x() , b=2, x x(-) , 则向量 \uf02bab ( ) 2. 一质点受到平面上的三个力123,,F F F (单位: 牛顿) 的作用而处于平衡状态. 已知1F ,2 F 成060 角, 且1F ,2 F 的大小分别为 2 和 4, 则3F 的大小为( ) ( A. 6 B. 2 C. 2 5 D. 2 7 \uf075\uf075\uf075 \uf0723. 设 P 是△ABC 所在平面内的一点,0PAPB\uf02b\uf03d B.4. 设向量a , b 满足: |公共点个数最多为 ( ) . A. 3 B. 4 C. 5 D. 6 5. 已知\uf028\uf0293,2 ,ab\uf03d \uf02d\uf03d \uf02d(A)72BC\uf072 C.| 4\uf03d,BABP\uf02b\uf03dPB\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072, 则( ) 0PC\uf03d D.. 以 a , b ,A.\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf0720PC| 3\uf03daPA, |\uf02b\uf03db\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf02b\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf0720PA\uf02db 的模为边长构成三角形, 则它的边与半径为1的圆的PBPC\uf02b\uf02b\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072\uf072 0\uf0d7 \uf03da ba\uf028\uf0291,0, 向量 a b\uf06c \uf02b与2ab\uf02d垂直, 则实数\uf06c 的值为( ) 1 1\uf02d (B)17 (C)16\uf02d (D)66. 8. 在平行四边形 ABCD中, AC 与 BD 交于点O EBD \uf03db , 则 AF \uf03d( ) 11\uf02bab B.33\uf072,是线段OD 的中点, AE 的延长线与CD 交于点 F . 若 AC \uf03d\uf075\uf075\uf075\uf072a ,\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072A.4221\uf02bab C.1124\uf02bab \uf072//b D、 ( 2, 4)\uf072\uf072与 a垂直, 则\uf06c 是( ) D.123\uf0723\uf02bab 7. 3. 已知平面向量A、 ( 5, 10)\uf02d \uf02d(1,2)\uf02d(1, 3)a \uf03da \uf03d,( 2,\uf03d \uf02d C、 ( 3, 6)\uf072)\uf02d \uf02dbm\uf072, 且a\uf072, 则 23ab\uf02d\uf02b\uf072=( ) B、 ( 4, 8)\uf072\uf02d\uf02d\uf02d8. 5. 已知平面向量A. \uf02d1 B. 1 C. \uf02d2 D. 2 9. 4. 若向量 , a b满足|| |a\uf03d122,(4, 2)\uf02db \uf03d,ab\uf06c \uf02b\uf072\uf072 \uf072| 1\uf03db\uf072\uf072, a\uf072与b\uf072的夹角为 60\uf0b0 , 则 a aa b\uf0d7 \uf02b \uf0d7 \uf03d\uf072 \uf072\uf072 \uf072( ) A. B.3 C.312\uf02b D. 2 10. 已知平面向量(1 1), ,(11)\uf03d\uf03d\uf02d,ab, 则向量1322\uf02d\uf03dab( ) A. ( 211. 在直角 ABC1)\uf044\uf02d\uf02d, B . ( 21)中, CD 是斜边 AB 上的高, 则下列等式不.成立的是( ) \uf02d , C . ( 1 0)\uf02d , D . ( 1 2)\uf02d , (A)2ACAC AB\uf0d7\uf03d\uf075\uf075\uf075\uf072\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072 (B) 2BCBA BC\uf0d7\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075 \uf072 (C)2ABAC CD\uf0d7\uf03d\uf075\uf075\uf075 \uf072\uf075\uf075\uf075\uf072 \uf075\uf075\uf075 \uf072 (D) 22() (\uf0b4\uf075\uf075\uf075 \uf072)AC AB\uf0d7BA BC\uf0d7CDAB\uf03d\uf075\uf075\uf075 \uf072 \uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072 12. 已知向量(1)( 1, )\uf03d \uf02dnn\uf03d,,ab, 若2 \uf02dab 与b 垂直, 则\uf03da( ) A. 1 B.二、 填空题 2 C. 2 D. 4 1. 若平面向量a , b 满足1\uf03d\uf02bba,ba \uf02b\uf072平行于 x轴,\uf072) 1, 2 ( \uf02d\uf072和向量b\uf072 . \uf03db, 则\uf072\uf03da . \uf072 \uf072= 2. 已知向量a\uf072和向量b\uf072\uf072的夹角为 30o, | | 2,| |3ab\uf03d\uf03d, 则向量a\uf072的数量积a b \uf0d73. 已知向量a\uf072和b的夹角为0120 , |\uf072| 1,|\uf03d, || 3\uf03da\uf075\uf075\uf075 \uf072ab\uf072\uf072, 则| 5\uf072|ab\uf02d\uf03d4. 已知向量(0, 1,1)\uf02da \uf03d\uf072,(4,1,0)b \uf03d|29b\uf06c \uf02b\uf03d\uf072且\uf075\uf075\uf075\uf0720\uf06c \uf03e, 则\uf06c = . abc5. 设 O、 A、 B、 C 为平面上四个点,\uf075\uf075\uf075 \uf072OA= a , OB= b , OC= c , 且 \uf02b \uf02b \uf03d 0,1a b\ue000b c\ue000c a\ue000\uf03d\uf03d\uf03d \uf02d, 则 abc\uf02b\uf02b=_______. 6. 在平行四边形 ABCD 中, AC 与 BD 交于点 O EBD \uf03db , 则 AF \uf03d______. (用 a,b 表示) (4cos ,sin ),a\uf061\uf061\uf03d,是线段OD 的中点,AE 的延长线与 CD 交于点 F . 若 AC \uf03d\uf075\uf075\uf075\uf072a ,\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf0727. 设向量(sin,4cos),(cos, 4sin\uf02d)bc\uf062\uf062\uf072\uf062\uf062\uf03d\uf03d\uf072\uf072\uf072\uf072,(1)若a\uf072与2bc\uf02d\uf072\uf072垂直, 求 tan()\uf061\uf062\uf02b值;(2)求||bc\uf02b\uf072\uf072的最大值; (3) 若 tantan16\uf061\uf062 \uf03d, 求证: a∥ b. . 8. 已知向量(cos ,sin )\uf071m\uf071\uf03d\uf075\uf072和\uf028\uf029\uf028\uf070\uf0292sin ,cos\uf071,,2n\uf071 \uf071\uf070\uf03d\uf02d\uf0ce\uf072, 且8 2,5mn\uf02b\uf03d\uf075\uf072\uf072求cos28\uf071\uf0e6\uf0e7\uf0e8\uf070\uf02b\uf0f6\uf0f7\uf0f8的值. 【体验练习题二: 】 一、 选择题: 1. 若向量 a =(1, 2), b =(1, -3), 则向量 a 与 b 的夹角等于( ) A 45 B 60 C 2. 在平面直角坐标系 xOy中作矩形 OABC , 已知\uf0b0\uf0b0\uf0b0120 D OA\uf0b0135 3\uf03d7\uf02d , 4\uf03dAB, 则· 的值为( ) A 0 B 7 C 25 D3. 向量, 的夹角为 120° , │ │ =│ │ =2, 则·(-) 等于( ) 324 \uf02b B 2 C 4. 已知向量a ≠ e , | e | =1, 对任意实数 t, 恒有| a -te | \uf0b3| a - e | , 则( ) A. a ⊥ e B. a ⊥(a - e ) C. e ⊥(a - e ) D. (a + e ) ⊥(a - e ) \uf028\uf029\uf028\uf0293,2 ,1,0ab\uf03d \uf02d\uf03d \uf02d\uf06c \uf02b1776. 已知向量(1,0),(0,1),(abckab k\uf03d\uf03d\uf03d\uf02b A.1k \uf03d 且c 与 d 同向 B. C.1k \uf03d \uf02d 且c 与 d 同向 D.k \uf03d \uf02d 且c 与 d 反向 7. 已知向量 a、 b 不共线, ck\uf03da\uf02b b(k \uf0ce R) , d\uf03da\uf02db, 如果 c // d, 那么( ) A.1k \uf03d 且 c 与 d 同向 B. C.1k \uf03d \uf02d 且 c 与 d 同向 D.二. 填空题: 8. 已知向量2 411\uf028\uf029\uf028\uf029, ,,a =b =. 若向量\uf05eb(1,2)OA \uf03d . 将OA为____________. 10. 设集合{\uf03dD平面向量} , 定义在 D 上的映射 f , 满足对任意 xA 324 \uf02d D 6 5. 已知, 向量 a b与2ab\uf02d垂直, 则实数\uf06c 的值为() 16ab\uf02d, 如果 //c d , 那么( ) k \uf03d 且c 与 d 反向 1 (A)\uf02d (B)1 (C)16\uf0ce\uf02d (D) ),1R d\uf03d1k \uf03d 且 c 与 d 反向 1k \uf03d \uf02d 且 c 与 d 反向 (\uf06c\uf075\uf075\uf075 \uf072)a + b , 则实数\uf06c 的值是 ; 绕着点 O 按逆时针方向旋转 90\uf06f得到向量 OB9. 设 O 为坐标原点, 向量 \uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072, 则 2OA OB\uf02b\uf075\uf075\uf075 \uf072\uf075\uf075\uf075 \uf072的坐标D\uf0ce, 均有 f (x) =\uf06c x( \uf06c \uf0ce R 且0\uf06c \uf0b9). 若\ufe31 a\ufe31 = \ufe31b\ufe31 且 a、 b 不共线, 则〔 f ( a) \uf02d f (b) 〕 \uf0d7(a+b) =________; 若11 . 若 把 函 数) 2(log2\uf02b\uf02d\uf03dxy为 . ) 8 , 4 (C),6 , 3 (B),2 , 1 (A, 且) 1\uf02b\uf0ae\uf02d\uf0ae\uf02d\uf03d ABBCf)(, 则\uf06c _______. 3的 图 象按 向 量 a 平 移 , 得到 函 数1(log2\uf02d\uf03dxy的 图 象, 则 向 量 a 的 坐 标12. 设向量,0),cos,(sin),0 , 1 (\uf070\uf0a3\uf071\uf0a3\uf071\uf071\uf03d\uf03dba则ba \uf02b的最大值为 _________. 13. 已知向量(1,0),(0,1),(),abckab kk \uf03d 且c 与 d 反向 1k \uf03d \uf02d 且c 与 d 反向 a\uf02b b(k \uf0ce R) , d\uf03da\uf02db, 如果 c // d, 那么( ) 1k \uf03d 且 c 与 d 反向 1k \uf03d \uf02d 且 c 与 d 反向 R dab\uf03d\uf03d\uf03d\uf02b\uf0ce\uf03d\uf02d, 如果 //c d , 那么( ) A. C.14. 已知向量 a、 b 不共线, c1k \uf03d 且 c 与 d 同向 B. C.1k \uf03d \uf02d 且 c 与 d 同向 D. 1k \uf03d 且c 与 d 同向 B.1k \uf03d \uf02d 且c 与 d 同向 D.1k\uf03d A.