ACJC 2025 Vectors I Lecture Notes

1 6A VECTORS IN TWO AND THREE DIMENSIONS, SCALAR AND VECTOR PRODUCTS SYLLABUS • Basic properties of vectors in two- and three-dimensions - 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 • Scalar and vector products in vectors - Concepts of scalar product and vector product of vectors and their properties - Angle between two vectors - Geometrical meanings of ˆa.n and ˆan , where ˆn is a unit vector

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Definitions and Notation ................................................................. 3 2 Simple Operations ........................................................................... 4 2.1 Addition of Vectors ................................................................ 4 2.2 Subtraction of Vectors ............................................................ 5 2.3 Scalar Multiple of a Vector .................................................... 5 3 Fundamental Results ....................................................................... 6 3.1 Parallel Vectors ...................................................................... 6 3.2 Collinear Points ....................................................................... 6 3.3 Linear Combinations of Vectors ............................................. 7 3.4 The Ratio Theorem ................................................................. 9 4 Two and Three Dimensional Vectors in Cartesian Form .............. 11 4.1 Vectors in Two Dimensions ................................................. 11 4.2 Vectors in Three Dimensions ................................................ 11 4.3 Operations on Vectors .......................................................... 12 4.4 Unit Vectors ......................................................................... 13 5 Scalar (Dot) Product ...................................................................... 15 5.1 Definition ............................................................................. 15 5.2 Properties of Scalar (Dot) Product ....................................... 15 5.3 Angle between Two Vectors ................................................ 16 5.4 Projection of a Vector in a Given Direction ........................ 19 6 Vector (Cross) Product .................................................................. 21 6.1 Definition ............................................................................. 21 6.2 Properties of Vector (Cross) Product .................................... 21 6.3 Perpendicular Distance from a Point to a Line .................... 23 6.4 Other Applications of the Vector Product ............................ 24 Annex A: Further Exploration: Applications of Vectors in Physics ....... 27 Annex B: Practice Questions on Vectors I ............................................. 28

6A Vectors I 3 LECTURE 1 Lesson Outline • Definitions and notation • Simple operations that involve addition, subtraction and scalar multiple of vectors • Fundamental results such as parallel vectors, collinear points, linear combinations of vectors and Ratio Theorem 1 DEFINITIONS AND NOTATION 1. A scalar is a quantity that is fully defined by magnitude alone. Examples: length, area, volume, distance, speed, temperature, energy, charge. 2. A vector is a quantity which has both magnitude and direction. Examples: displacement, velocity, acceleration, force, momentum 3. The vector a in the diagram goes from P to Q. (i) The direction of the vector in the diagram is denoted by the arrowhead. (ii) The vector can be denoted as PQ , or simply as a. In printed text, vectors are indicated by bold text. In writing, we indicate vectors with a wavy line underneath the letter: a . (iii) The position vector of a point P is the vector with reference to a fixed point O (origin) i.e. position vector of P is vector OP . (iv) A free vector is a vector of which only the magnitude and direction are specified but not the position. (v) A point moves from P to Q. The displacement vector PQ is a vector whose length is the shortest distance from the initial position P to the final position Q of the point. 4. The magnitude of PQ is the length of the line segment PQ, denoted as PQ, PQ or a . 5. Two vectors u and v are equal if and only if they have the same magnitude and direction, i.e. =  =u v u v and u and v are parallel and in the same direction. Practice: Draw another vector that is equal to a on the diagram. a P Q O

ACJC 2025/26 H2 Mathematics (9758) 4 6. Negative of a vector If two vectors u and v have the same magnitude but opposi te directions, then we say that =−vu , i.e., −u is a vector in a direction opposite to u and of the same magnitude as u. Practice: Draw −a on the diagram. 7. The zero vector or the null vector has zero magnitude. Geometrically, it is a point. It is denoted by 0. 2 SIMPLE OPERATIONS 2.1 Addition of Vectors We can add vectors using either the Triangle Law or the Parallelogram Law. Given vectors a and b, their sum =+c a b is defined as shown in the diagrams. Note that, in general, = +  +c a b a b . Equality holds only in the special case where a and b are parallel and in the same direction. In general, +c a b (the triangle inequality) To add three or more vectors, we can use the polygon law of addition. + + =a b c d Note Vector addition is commutative and associative, i.e., (Commutative) ( ) ( ) (Associative) + = + + + = + + a b b a a b c a b c a b a b c c a b c d

6A Vectors I 5 2.2 Subtraction of Vectors Given vectors a and b, the vector −ab is defined as ()+−ab , as shown in the diagram. Note If a and b are the position vectors of points A and B respectively, we can write the vector AB in terms of the vectors a and b. AB AO OB OA OB OB OA = + =− + = − = − ba 2.3 Scalar Multiple of a Vector If  is a positive real number, then a is a vector with magnitude  a in the direction of a. If  is a negative real number, then a is a vector with magnitude  a in the opposite direction to a. b a b a O A B a 2a

ACJC 2025/26 H2 Mathematics (9758) 6 3 FUNDAMENTAL RESULTS 3.1 Parallel Vectors If a and b are non-zero vectors, then a and b are parallel if and only if k=ba for some ,0kk . Note • If a and b are parallel and in the same direction, then k is positive. • If a and b are parallel and in opposite directions, then k is negative. • If =ab and a is not parallel to b, then it must be the case that ==a b 0 , so 0== . Example 1 ABCD is a quadrilateral and P is any point on AD. If ,AP PB PD PC+ + = prove that ABCD is a parallelogram. Solution AP PB PD PC AP PB PC PD DP PC AB DC + + =  + = − = + = Therefore AB and DC have the same direction and length. Hence ABCD is a parallelogram. ■ 3.2 Collinear Points When 3 points A, B and C lie on the same straight line, we say that A, B and C are collinear. From the diagram, since //AB AC , i.e. AB k AC= for some real value of k, and there is a common point A, therefore A, B, C are collinear. A, B and C are collinear, if and only if AB k AC= for some real value of k , 0k  , where A is a common point. B A C A B C D

6A Vectors I 7 Example 2 If O, P, Q and R are four points such that 10OP= a , 5OQ= b and 43OR=+ ab , show that points P, Q and R are collinear. Solution =−PQ OQ OP ( )5 10 5 2b a b a= − = − =−QR OR OQ ( ) ( )4 3 5 4 2 2 2a b b a b b a= + − = − =− − 5 2 =−PQ QR with a common point Q, P, Q and R a