Vectors ( 向量 ) Chapter 16 2D Vectors A vector is a line which has both magnitde and direction. For example, in a weather report yo may hear a statement like the wind is blowing at 25 knots ( 海浬 ) in the sotheast direction. This describes a vector the velocity of the wind. It has magnitde: 25 knots, and direction: sotheast. The qantities that are not vectors are called scalars ( 純量 ). Vectors Scalars Displacement ( 位移 ) Distant ( 距離 ) Velocity ( 速度 ) Speed ( 速率 ) Acceleration ( 加速度 ) Mass Force Energy Electric Field ( 電場 ) Electric Charge ( 電荷 ) Representation of Vectors We represent a vector by an arrow, like this: While a vector is connecting two points, we call that a directed line segment ( 有向線段 ) P In the above figre, a vector is connected from P to Q. We say that P is the initial point, and Q is the terminal point, as the vector starts from P and ends at Q. This vector is written as PQ or PQ. To distingish vectors from highlighted words, we se the former notation here. It is not necessary to emphasize where the vector starts and ends. We may let the vector to have its specific name: This vector is called. It may also be written as or. To denote the magnitde of a vector, we write. Q Special Vectors Eqal vectors The length of the arrow describes its magnitde, while the direction the arrow head pointing at shows its direction. Two vectors are eqal if both of them posses the same magnitde and direction. Ths the following two vectors are eqal, althogh they have different start and terminal points. 122
Zero Vector ( 零向量 ) Zero vector is vector which the magnitde is 0. It is denoted as 0, 0 or 0. A zero vector has no specified direction. Unit Vector ( 單位向量 ) Unit vectors are vectors which the magnitde is one nit, i.e., if a = 1, then a is a nit vector. To give stress on a nit vector, it is sally written with a ^ sign, like â. Negative Vector ( 負向量 ) A negative vector of a vector is a vector which has same magnitde of bt opposite direction. It is denoted as. Vector Operations Addition Two vectors are added together by connecting initial and terminal points. This is called the triangle law of addition ( 三角形加法 ) AC = AB+ BC AC AB BC + v v Besides, the sm can also be the diagonal of the parallelogram made by the two vectors concerning. This is called the parallelogram law of addition ( 平行四邊形加法 ). If several vectors are added together, pt them in order connecting the tails to the heads, forming a polygon. Then the sm is the initial point of the first vector connecting to the terminal point of the last vector: a+ b+ c+ d + e+ f + g+ h Or algebraically: n 1 A A = A A k k+ 1 1 n k= 1 This is the polygon law of addition ( 多邊形加法 ). Sbtraction + v = + v g Scalar Mltiplication v g a + v f e When a vector v is mltiplied by a scalar k, it means to enlarge v by a factor of k. b d c 123
v 2v 3v 1 2 v Any vectors can actally be represented by position vectors. For example: PQ = OQ OP Point of division If on a line segment AB, a point P divides it by a ratio of r: s, then sa+ rb p= s+ r A P Properties of Vector Operations 1. + v = v+ 2. + ( v + w ) = ( + v ) + w m n = m = n m 4. m( + v) = m+ mv m+ n = m+ n 3. 5. (Commtation of Addition) (Association of Addition) (Association of Scalar Mltiplication) (Distribtion of Addition) (Distribtion of Scalar Mltiplication) a p B b O Components of Vectors and Vectors in Coordinates System The components ( 分向量 ) of a vector are vectors of a and b sch that a + b =. Usefl Theorems If and v are not parallel (their directions are not the same): m+ nv = 0 m= 0, n= 0 m+ nv = p+ qv m= p, n= q Position Vectors ( 位置向量 ) Position vectors are vectors from a fixed reference point (origin) O to a specific point. R a b a b a P Q Origin, O S b 124
With so many ways to resolve a vector into components, we often resolve it into two perpendiclar components, like the right-most figre in the last page. yj O Y (0, y) P (x, y) θ X (x, 0) xi The idea to represent a vector in a coordinates system is to resolve it into components, which are parallel to the x- and y-axis respectively. Let be the vector OP, which the coordinates of P are (x, y). So OY is the projection ( 投影 ) of onto the y-axis, and OX is onto the x-axis. Bt it is rather inconvenient to write the components this way. So we introdce the nit base vectors ( 基本單位向量 ): i : The nit vector connecting O and (0, 1). I.e., the nit vector of the x-axis. j : The nit vector connecting O and (1, 0). I.e., the nit vector of the y-axis. Ths the two components can be represented by yj and xi respectively. And the vector can then be written as: = xi + yj Any vectors on the coordinates plane can be written in terms of i and j. Moreover, 2 2 = x + y 1 y θ = tan x Where θ is the angle made between and the x-axis (The direction ). Scalar Prodct of Two Vectors We have seen that a vector and a scalar can be mltiplied together. Bt how abot two vectors? Yes, there are ways to mltiply two vectors together. Actally, there are two types of mltiplications. We will introdce the scalar prodct ( 純量積 ) here first. (The other will be taght in Chapter 19) The prodct is defined in this way: The scalar prodct of a and b is the magnitde of a mltiplied by the projection of b onto a. Written explicitly: a b = a b cosθ Where θ is the angle between the two vectors. The reslt of the scalar prodct is a scalar, and ths its name. Moreover, the scalar prodct is sometimes called the dot prodct ( 點積 ), as its operator is a dot. Note that this dot can never be omitted. Properties of Scalar Prodct 1. a b = b a 2. a ( b+ c) = a b+ a c ka b = k a b = a kb a // b a b = a b 3. 4. 2 5. a a = a 6. If a, b 0 7. a b a b, ( a b) ( a b = 0) (Commtative) (Distribtive with Addition) (Distribtive with Scalar Mltiplication) (Parallel Vectors) (Self-Operation) θ (Perpendiclar Vectors) (Ineqality) Scalar Prodct on Coordinates Plane If a = x1i + y1 j, b = x2i + y2 j, then a b = x1x 2+ y1 y2. b a 125
Example: Prove that three altitdes of a triangle intersect at one point Applications of Vectors We sally se vectors in proving, especially for: Parallel lines Collinear points ( 共線點 ) Perpendiclar lines Finding ratio of line segments (This is not proving, indeed) Angle between two lines Etc, etc, etc Since two altitdes always intersect at one point, we need to prove the third also pass throgh that point. Conversingly, we may prove that the line (vector) connecting that point and the 3 rd vertex is perpendiclar to its corresponding side. A F H E Example: Assme AB is a chord of the circle O. M is the mid-point of AB. Prove that AB OM. To prove AB OM, we will show AB OM = 0. So let a = OA, b = OB, m= OM. We have a = b. A Moreover, AB = b a. Last bt not least, since M is the m = 1 a + b. mid-point of AB, it means that Ths: 1 AB OM = ( b a) 2( a+ b) = 1 b b a a a b+ a b 2 1 2 2 = b a 2 = 0 Which confirms or gess. 2 M O B B D C Let b = CB, h = CH, a = CA. Ths AB= b a, BH = h b, AH = h a. Since BH AC, we have: Also, as AH BC, ( h b) a = 0 a h a b = 0 1 ( h a) b = 0 b h a b = 0 2 (2) (1): b h a h = 0 ( b a) h = 0 AB CH = 0 Ths CH AB. If CH is prodced to meet AB at F, then CF AB, which means CF is yet another altitde. Therefore, the three altitdes meet at one point. 126
Revision In this chapter, we ve learnt: 1. What is vector 2. Simple operations on vectors 3. Position vectors 4. Representing a vector in coordinate system 5. Scalar prodct 6. Application of vectors Exercise In the followings, if not specified, x is the variable. Denote the statement for M.I. as P or S. 1. Prove the polygon law of addition in page 123 sing M.I. 2. Prove the eqation of scalar prodct on coordinates plane in page 125 sing the standard definition of scalar prodct. 3. If G is the centroid of ABC, show that AG+ BG+ CG = 0. 4. Assme ABCD is a qadrilateral. Let P, Q, R, S be midpoints of AB, BC, CD and DA respectively. Show that PQRS is a parallelogram. 5. Prove cosine law sing vector method. 6. (HKCEE 1994) P, Q and R are points on a plane sch that OP = i + 2 j, OQ= 3i + j and PR = 3i 2 j, where O is the origin. a) Find PQ b) Find PQ c) Find cos QPR. 7. If O and H are respectively the circmcenter and orthocenter of ABC, prove that OH = OA+ OB+ OC. 8. (BkMO 2) Let O be the circle throgh points A, B, C and let D be the mid-point of AB. Let E be the centroid of ACD. Prove that CD OE iff AB = AC. 9. Try to prove all the theorems listed in Chapter 9 by vector method. Some may not be able to done in this way, so try yor best. 127
Sggested Soltions for the Exercise 6a) 2i j b) 5 4 65 c) 65 128