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1、高中数学必修4平面向量知识点与典型例题总结(生)高中数学必修4平面向量知识点与典型例题总结(生) 1, mathematics will be the basic type of questions - plane vector basic concepts and formulas any time to write the vector should be accompanied by arrows) 1. vector f has both magnitude and direction of. As f or AB A. Die 2. the size of the K vector
2、vector from La in length or denoted as f | |AB or | | A. The vector f length of 3. units of 1 vectors. If e is a unit vector in it | | in 1E. 4. zero vector k vector of length 0. As f 0. 0 direction is arbitrary and arbitrary vector in parallel 5. parallel vector vector from collinear vectors shown
3、the same or opposite direction F. 6. vector k equal vector length and direction are the same. Vector of 7. opposite vectors of equal length in the opposite direction of the f. AB BA rate. 8. f AB BC AC triangle law In the AB BC CD DE AE will share in his AB AC CB will share in the rate in those to i
4、ncrease the minuend 9. parallelogram rule f In order to, A B is the two diagonal parallelogram shaped edges were a B in a in the B rate. 10. / F / collinear theorem A B a B in chemical exergy. When the end of 0 when B and a included the same direction when the 0 share in a B and exergy exergy when r
5、everse. Two vectors of 11. basal K arbitrary collinear a group known as the base. If the 12. K vector mode (a, x) y in 2 2| | A x y in 2 in the end 2| | A a 2| (a) | in B a B in his will 13. the number of product and angle formula F | | | |cosa B a B from La La will share cos | | | | A B A B from La
6、 In La 14. parallel and vertical R 1221 /a B a B x / y x y in the end of 12120 share in exergy in the 0A B a B x x y y in La in the end of his will 1. types of basic concepts of judge f From the 1 observed collinear vectors is in the same line vector. From the 2 observed that if the two vector is no
7、t equal in their end point may not be the same. 3 compared with known unit vector from collinear vector is only. From the 4 ABCD La quadrilateral is a parallelogram condition is AB CD In. From the 5 increase if AB CD In the end, A, B, C, D four points form a parallelogram. From the 6 increase becaus
8、e the vector is directed line segment in the axis vector is so. If those 7 increase A and B in B and C in the collinear collinear A and C collinear. From the 8 increase if Ma MB In the end, a B. 2 of the 9 if Ma Na is m n exergy exergy exergy. The 10 if A and B A and B are not collinear exergy are n
9、ot zero vector. The 11 if | | | |a B a B / / exergy exergy A B. If the 12 | | | | A B a B in a B the rate of exergy exergy. Question 2. vector addition and subtraction 1. A says, go east, 8km, and B says go north, 6km |a B in the | exergy . 2. simplify () () AB MB BO BC OM in the in the in the in th
10、e exergy. 3. known | | 5OA | | exergy exergy, 3OB, |AB, | the maximum value and minimum value respectively. ,. 4. known AC, AB, AD With the vector and the exergy and AC a, BD B AB exergy exergy exergy exergy Exergy exergy AD. The 5. known points on segment AB and C included 3 Five AC AB AC BC exergy
11、 exergy, exergy exergy of AB BC. Number multiplication of 3. vectors of a problem 1. from the 1 Calculation 3 (2) (a) B a B in the 2 rate in the exergy 2 (253) 3 (232) A B C a B C in the rate rate rate rate in exergy 2. known (1, 4), (3,8) a B 1 rate exergy exergy exergy rate Three Two The a B rate
12、exergy . Question 4., graphing, sum of spherical vectors Given vector, A B if you make the exergy exergy vector 1 Three Two A and B in 32 Two A B rate. A B Question 5.: seek unknown vectors by a given vector according to a graph 1. known in ABC D is in use in exergy vector ABAC AD BC said the midpoi
13、nt of exergy exergy. In 2. the parallelogram ABCD included known AC a, BD B and AB AD exergy exergy exergy And. Coordinate calculation of the 6. vectors of the questions 1. known (4,5) AB (2,3) exergy exergy A is the B coordinate is included. 2. known (3, 5) PQ exergy exergy rate rate (3,7) P is the
14、 Q coordinate is included. 3. if the object is subjected to three forces, 1 (1,2) F 2 (2,3) F exergy and exergy rate, 3 (1, 4) F rate is the rate exergy, force coordinates . 34. known (3,4) a (5,2) B in exergy rate in a B in the exergy rate in a B in 3 2A B rate. 5. known (1,2), (3,2) A, B, vector (
15、2, 32) a x x y in the AB rate and exergy rate will equal to, The value of X y. 6. known (2,3) Exergy in AB (m, BC) n (1,4) CD in exergy exergy rate in DA exergy . The 7. known O is the origin of coordinates in (2, 1), (4,8) Coordinate A and the 3 0AB rate B rate in BC in OC in the exergy. The questi
16、on 7. determines whether the two vectors can serve as a set of bases 1. known 12, E e is a group of basal plane will determine whether each of the following vector can constitute a group of base A.1 212 E e e e in B.1 e e 22132 6e rate and e rate rate and 4 C.1 2213 3E e e e D.2 1E E 2 in the rate a
17、nd e rate and 2. known (3,4) A and a can form the basement in exergy exergy rate is A.3 4 (,) 55 B.4 3 (,) 55 C.3 4 (,) 55 Rate rate D.4 (1) Three Occlusion rate Question 8. combines trigonometric function to find vector coordinates 1. O is known in origin of coordinates in the second quadrant in A
18、In 150 2OA | | exergy The coordinates xOA and OA in exergy in. The 2. known O is the origin in point in the first quadrant in A 4 3OA in 60 | | exergy The coordinates xOA and OA in exergy in. Question 9., find quantity product 1. known | | 3, | | 4A B and a B and exergy exergy in the angle of 60 x 1
19、 a B rate for exergy The rate in 2 exergy (a) a in the B in 3 exergy rate 1 () Two A B B in the 4 exergy occlusion rate (2) (3) a B a B rate in the. 2. known (2, 6), (8,10) The a B rate in rate for exergy exergy exergy rate 1 | |, | |a B in a B 2 exergy rate 3 (2) to the b 4 (2) (3) (B to b . 题型10.求
20、向量的夹角 1.已知 | | 8, | | 3a b 12 The b 求a与b的夹角. 2.已知 (3.1), 2 (3.2) b 求a与b的夹角. 3.已知 (1,0) A (0,1) b (2,5) c 求cosbac . 4 题型11.求向量的模 1.已知 | | 3, | | 4A b 且a与b的夹角为60 求 1 | | The b 2 | 2 3 | the b . 2.已知 (2, 6), (8,10) The b 求 1 | |, | | the b 5 | | The b 6 1 | | 2 The b . 3.已知 | | 1 | | 2A b | 3 2 | 3 The
21、 b 求 | 3 | the b . 题型12.求单位向量 【与 A平行的单位向量 | | The And The 】 1.与 (12.5) A 平行的单位向量是. 2.与1 (1) 2 M 平行的单位向量是 . 题型13.向量的平行与垂直 1.已知 (6.2) a (3) (B m 当m为何值时 1 / / The b 2 a b 2.已知 (1,2) A (3.2) b 1 K为何值时 向量ka b 与3a b 垂直 2 K为何值时 向量ka b 与3a b 平行 3.已知 A是非零向量 a B c 且b c 求证 () (B c . 题型14.三点共线问题 1.已知 (0, 2) A (
22、2,2) b (3,4) c 求证 , B c三点共线. 2.设2 (5), 2, 8, (3) 2 AB a B CB B CD b 求证 a B D. 、 、三点共线. 5 3.已知2, 5, 6, 7 2Ab B CB B CD b 则一定共线的三点是. 4.已知 (1, 3) A (8, 1) b 若点 2 (1, 2) C a 在直线ab上 求a的值. 5.已知四个点的坐标 (0,0) O (3,4) a (1,2) b (1,1) c 是否存在常数t 使oa tob oc 成 立 题型15.判断多边形的形状 1.若3 Ab e 5cd e 且 | | | | ad bc , 则四边
23、形的形状是 . 2.已知 (1,0) A (4,3) b (2,4) c D 证明四边形abcd是梯形 (0.2). 3.已知 (2,1) A (6, 3) b (0.5) c 求证 abc 是直角三角形. 4.在平面直角坐标系内 (1.8), (4.1), (1.3) OA ob oc , 求证 abc 是等腰直角三角形. 题型16.平面向量的综合应用 1.已知 (1) 0) Exergy in a (2,1) B in the exergy Why K value in Ka B and 3a B rate vector in parallel calculation 2. known (
24、3, 5) a and a B in exergy In the | | 2b in B coordinates for exergy. 3. known a B And in the same direction (1,2) B in 10 exergy A B in a coordinates for exergy. 3. known (1,2) Exergy in a (3,1) B (5,4) in exergy C C A in the exergy exergy in B. 4. known (5,10) Exergy in a (3, 4) B (5,0) in exergy o
25、cclusion rate Please use the vector C in exergy A, B stands for vector C. 5. known (3) A m in exergy (2, 1) B rate in 1 if the exergy exergy rate The range of a and B angle is an obtuse in M 2 if the exergy rate The range of a and B for acute angle in M. 6. known (6,2) a (3, b) in exergy exergy rate
26、 in m when the m value in 1 why rate a and B exergy exergy rate angle is an obtuse angle angle of 2 A and B of the exergy Included angle 7. the vertex coordinates of the known trapezoidal ABCD are (1,2) The A rate in (3,4) B (2,1) D in DC / /AB in 2AB and CD in exergy Find the coordinates of the poi
27、nt C. The coordinates of 68. known parallelogram three vertices form ABCD respectively (2,1) A (1,3) B increase rate increase (3,4) C coordinates for the fourth vertex of D la. 9. I ship with the speed of 5km/h perpendicular to the direction of travel across the sailing direction and increase ship f
28、low direction angle is observed for 30 Velocity of water and actual speed of a ship. 10. known ABC The coordinates in three vertices respectively (3,4) A (0,0) B (la la la, 0) C C 1 if those 0 included AB AC la la Exergy for the value of C from 2 K exergy if 5C exergy is observed for the value of si
29、nA. spare The 1. known | | | | 3, 4, 5 | | A B a B in LA for exergy exergy exergy |a B and | rate vector, The angle between a and B. 2. known x, a, B Exergy in La 2Y a B in La | | | exergy and exergy exergy | 1A B La a B For LA The cosine of the angle between the X and y. 1. known (1,3), (2, 1) A B
30、is the rate increase exergy exergy rate (32) (25) a B a B in the exergy rate increase . 4. known two vectors (3,4), (2, 1) A B A is observed when exergy exergy rate of XB a in the B rate and vertical when the value of X. 5. known two vectors (1,3), (2,) A B La a B and exergy exergy exergy angle from
31、 angle of Exergy for la. 1、20以内退位减法。If the variation in the (, 2), (3,5) a B a B La exergy exergy exergy rate From the angle of the range and increase the Exergy for an. Special methods, fill in the end 同心圆：圆心相同，半径不等的两个圆叫做同心圆。1. special case method In the cases of goods in P27 4. Because of M, N in
32、AB, any location on the AC have set up special circumstances so La La M, N and B, C 1. 仰角:当从低处观测高处的目标时，视线与水平线所成的锐角称为仰角La can get 1 overlap tanA不表示“tan”乘以“A”；M n 2m n in the La exergy exergy exergy. 2. substitution validation A、当a0时In the case of known vector (1,1), (1, 1), (1, 2) 最值：若a0，则当x=时，；若a0，则
33、当x=时，A B C exergy exergy exergy rate rate is C rate increase from exergy exergy A.1 3 2. 俯角：当从高处观测低处的目标时，视线与水平线所成的锐角称为俯角22 A B B.1 occlusion rate of 3 22 The a B rate in C.3 1 22 A B rate D.3 1 22 定义：在RtABC中，锐角A的对边与斜边的比叫做A的正弦，记作sinA，即;The a rate in B (3) 扇形的面积公式:扇形的面积 (R表示圆的半径, n表示弧所对的圆心角的度数)In the
34、known variant (1,2), (1,3), (1,2) A B C exergy exergy exergy rate increase rate with a, said B C. 3. exclusion In the case of known M is ABC In the center of La is the following vector and AB collinear (3)相离: 直线和圆没有公共点时,叫做直线和圆相离.is from exergy A.AM MB BC In the B.3AM AC in C.AB BC AC in D.AM BM CM in his in his