2026 Chp 1A (Student) - JPJC

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 1 Chapter 1: Vectors Content Outline I. Basic properties of vectors in two- and three dimensions Include: • addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations • position vectors, displacement vectors and direction vectors • magnitude of a vector • unit vectors • distance between two points • collinearity • use of the ratio theorem in geometrical applications II. Scalar and vector products in vectors Include: • concepts of scalar product and vector product of vectors and their properties • calculation of the magnitude of a vector and the angle between two vectors • geometrical meanings of ˆan  and ˆan , where ˆn is a unit vector Exclude triple products a • b × c and a × b × c. III. Three-dimensional vector geometry Include: • vector and cartesian equations of lines and planes • finding the foot of the perpendicular and distance from a point to a line or to a plane • finding the angle between two lines, between a line and a plane, or between two planes • relationships between (i) two lines (coplanar or skew) (ii) a line and a plane (iii) two planes Exclude: • finding the shortest distance between two skew lines • finding an equation for the common perpendicular to two skew lines

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 2 References • Websites (1) http://www.h2maths.site (2) https://www.mathsisfun.com/algebra/vectors.html • Books (1) Ho Soo Thong, Tay Yong Chiang & Koh Khee Meng, “College Mathematics Syllabus C Volume 1”, Pan Pacific Publications, Call Number: HO510 (2) Pure Mathematics by Alan Sherlock, Elizabeth Roebuck, Timothy Heneage, Shirley Beck: Chapter 16 (3) Pure Mathematics 2 by L Bostock, S Chandler: Chapter 5 (4) My First Step in using TI-84 Plus CE for H1 and H2 Math by Attal Lam Chapter 1(a): Vector Algebra and Scalar Product 1. Introduction Pre-requisite: GCE “O” Level Elementary Mathematics – 2-dimensional vectors • A scalar quantity is one that is fully defined by magnitude alone, e.g. length, distance, speed. • A vector quantity is one that has both magnitude and direction, e.g. force, displacement, velocity. Geometrical Representation A vector a (directed from P to Q) may be represented geometrically by a directed line PQ . Notation: , or PQ a a   (in the direction from P to Q) 2. 2-Dimensional / 3-Dimensional Vectors Examples of 2D vectors: 1 1   , 4 2 −  . Examples of 3D vectors: 1 1 4    , 4 2 0 −   . Column vector form Q P a 

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 3 Geometrical Representation of 2D Vectors 2D vectors can be represented on the Cartesian plane with the i value corresponding to the x-coordinate and the j value corresponding to the y-coordinate. The zero vector corresponds to the origin (0, 0), denoted by 0 or 0  . 10,01    ==       ij For example, OA =  5i + 2j = 5 2   Magnitude/modulus of OA  (= OA  = length of vector OA  = OA) = 225 2 29+= (by Pythagoras theorem) In general, the magnitude/modulus of a vector i + j = xPQ x y y =   is 22PQ x y=+  Geometrical Representation of 3D Vectors 3D vectors can be represented on the Cartesian space with the i value corresponding to the x-coordinate, the j value corresponding to the y-coordinate and the k value corresponding to the z-coordinate. 0 or 0  corresponds to the origin (0, 0, 0). 1 0 0 0 , 1 , 0 0 0 1            = = =                i j k For example, OB =  10i + 4j + 5k = 10 4 5    Magnitude/modulus of OB  (= OB  = length of vector OB  is OB) = 2 2 210 4 5++ = 141 In general, the magnitude/modulus of a vector x PQ x y z y z  =  i + j+ k =  is 2 2 2PQ x y z= + +  Recall : In Chp 0A, | x | is the distance of the real number x from zero on a number line. 4 5 10 B(10, 4, 5) O z-axis y-axis x-axis y x O 2 5 A (5, 2) j i i k j

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 4 Example 1 Find the magnitude of the following vectors, in exact form: (i) 34OA =+ij  , (ii) 6 3 4OB =+ i j+ k  , (iii) 1 1 2 OC  = −  Solution (i) ( ) ( ) 22 34OA =+  = 5 (ii) OB  = ( ) ( ) ( ) 2 2 2 634++ = 61 (iii) OC  = ( ) ( ) ( ) 2 2 2 1 1 2 6+ + − = 3. General types of vectors 3.1 Equal Vectors 3.2 Negative Vectors 3.3 Parallel Vectors 3.4 Unit Vectors 3.5 Position Vectors 3.6 Displacement Vectors 3.1 Equal Vectors Two vectors are equal if and only if they have the same magnitude and direction. Same magnitude, same direction, so equal Different magnitude, same direction, so not equal Same magnitude, different direction, so not equal i.e. == a b a b AND and are in the same directiona b

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 5 P Q P Q 3.2 Negative Vectors If two vectors a and b have the same magnitude but are parallel in opposite directions, then a = –b or –a = b. From the diagram, we can see that the directed line segments PQ and QP  have the same magnitude but are parallel in opposite directions. PQ QP = −  Given that ABCD is a parallelogram, state all the pairs of equal vectors. AB DC=  and BCAD = Do you know why AD CB  and AB CD  ? 3.3 Parallel (//) Vectors If and ba are non-zero vectors, then // for some  = a b a b  where 0  . In other words, if a and b are any two vectors and =ab for some    , 0  , then there are 2 possibilities . Either (i) a and b are parallel, (ii) a = b = 0 Same direction, parallel (e.g. 2=ab ) Opposite direction, parallel (e.g. 2=−ab ) Different direction, not parallel A B C D Note: To prove ABCD is a parallelogram, we need to prove either AB DC=  OR AD BC=  . b a b a

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 6 Example 2 Determine if the following pairs of vectors are parallel (i) 4i + 4j – 2k and –2i – 2j + k, (ii) 3i – 5k and 0 3 5  − . Solution (i) Since ( ) 42 4 2 2 21 −       = − −      −     they are ___________. (ii) There exist ______________ such that 3 0 5   − =  0 3 5  −  they are ______________. Example 3 Given that 2i + hj + 4k is parallel to 5i – 4j + pk, find the values of h and p. Solution Since 25 is parallel to 4 4 h p         −          , 25 4 4 h p         =−          for some wher 0. e     2 5 (1) 4 (2) 4 (3) h p    =− = − − =− 2From (1), 5 = From (2), 284 55h = − = − From (3), 24 10 5 pp=  = Note: Observe that in (ii), when we try to solve for  , there is no unique  value found.

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(a) Vectors − Vector Algebra & Scalar Product / Pg 7 3.4 Unit Vector The unit vector of a , denoted by ˆa , is the vector in the direction