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Unformatted text preview: Vectors Advanced Level Pure Mathematics Advanced Level Pure Mathematics Algebra Chapter 7 Vectors 7.1 Fundamental Concepts 2 7.2 Addition and Subtraction of Vectors 2 7.3 Scalar Multiplication 3 7.4 Vectors in Three Dimensions 5 7.5 Linear Combination and Linear Independence 7 7.6 Products of Two Vectors 13 A. Scalar Product B. Vector Product 7.7 Scalar Triple Product 22 Matrix Transformation* 24 7.1 Fundamental Concepts 1. Scalar quantities: mass, density, area, time, potential, temperature, speed, work, etc. 2. Vectors are physical quantities which have the property of directions and magnitude. e.g. Velocity v , weight w , force f , etc. 3. Properties: Prepared by K. F. Ngai Page 1 7 Vectors Advanced Level Pure Mathematics (a) The magnitude of u is denoted by u . (b) CD AB = if and only if CD AB = , and AB and CD has the same direction. (c) BA AB- = (d) Null vector, zero vector , is a vector with zero magnitude i.e. = . The direction of a zero vector is indetermine. (e) Unit vector, u ˆ or u e , is a vector with magnitude of 1 unit. I.e. 1 = u . (f) u u u = ˆ ⇔ u u u ˆ = 7.2 Addition and Subtraction of Vectors 1. Geometric meaning of addition and subtraction. AD CD BC AB = + + p q PQ- = 2. Properties: For any vectors v u , and w , we have (a) u v v u + = + , (b) w v u w v u + + = + + ) ( ) ( , (c) u u + = + (d) ) ( ) ( = +- =- + u u u u N.B. (1) ) ( v u v u- + =- Prepared by K. F. Ngai Page 2 Vectors Advanced Level Pure Mathematics (2) b c a b a c- = ⇒ + = 7.3 Scalar Multiplication When a vector a is multiplied by a scalar m, the product ma is a vector parallel to a such that (a) The magnitude of ma is m times that of a . (b) When m , ma has the same direction as that of a , When < m , ma has the opposite direction as that of a . These properties are illustrated in Figure. Theorem Properties of Scalar Multiplication Let n m , be two scalars. For any two vectors a and b , we have (a) a mn na m ) ( ) ( = (b) na ma a n m + = + ) ( (c) mb ma b a m + = + ) ( (d) a a = 1 (e) o a = (f) = α Theorem Section Formula Let A,B and R be three collinear points. If n m RB AR = , then n m OA n OB m OR + + = . Example Prove that the diagonals of a parallelogram bisect each other. Solution Prepared by K. F. Ngai Page 3 Vectors Advanced Level Pure Mathematics Properties (a) If b a , are two non-zero vectors, then b a // if and only if mb a = for some R m ∈ . (b) b a b a + ≤ + , and b a b a- ≤- 7.4 Vectors in Three Dimensions (a) We define k j i , , are vectors joining the origin O to the points ) , , 1 ( , ) , 1 , ( , ) 1 , , ( respectively. (b) j i , and k are unit vectors. i.e. 1 = = = k j i . (c) To each point ) , , ( c b a P in 3 R , there corresponds uniquely a vector ck bj ai p OP + + = = where is called the position vector of P ....
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