NJC Vectors 1A

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Notes Page 1 of 35 National Junior College 2025 – 2026 H2 Mathematics Topic 1A Vectors I Notes §1 Introduction Key Questions: ☐ What are scalars and vectors? ☐ How can vectors be represented geometrically? ☐ What is the magnitude of a vector? ☐ What is a zero vector? ☐ What is a unit vector? ☐ How do we find the unit vector of a given vector? 1.1 Scalars and Vectors What is a scalar? Definition 1.1 (Scalar) A scalar is a quantity with magnitude but no direction. Examples of scalars Mass, distance, speed What is a vector? Definition 1.2 (Vector) A vector is a quantity with both magnitude and direction. Examples of vectors Force, displacement, velocity 1.2 Geometric Representations of Vectors How can vectors be represented geometrically? Geometrically, a vector can be represented by a directed line segment, where the arrowhead represents the direction of the vector. The points P and Q are called the start and end points of the vector PQ ⎯⎯ → respectively. How are vectors denoted? PQ ⎯⎯ → , u (in print form), u (in written form)

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Notes Page 2 of 35 1.3 Magnitude of a Vector What is the magnitude of a vector? Definition 1.3 (Magnitude) The magnitude of a vector u is the length of the line segment that it is represented by u . \ How is the magnitude of a vector denoted? PQ ⎯⎯ → , u (in print form), u (in written form) 1.4 Zero Vectors and Unit Vectors What is a zero vector? Definition 1.4 (Zero Vector) A zero vector is a vector whose magnitude is zero. How is a zero vector denoted? 0 (in print form), 0 (in written form) What is a unit vector? Definition 1.5 (Unit Vector) A vector with magnitude 1 is called a unit vector. The unit vector of a given vector u is the vector with the same direction as u and magnitude 1. How is the unit vector of a given vector denoted? The unit vector of a given vector u is denoted by ˆu (in print form) and ˆu (in written form). How do we find the unit vector of a given vector? Example 1.1 Given that the vector a has magnitude 3, express the unit vector of a in the form ka for some scalar k. Divide the vector by its magnitude. Solution Since the vector a has magnitude 3, one -third of a has magnitude 1 and shares the same direction as a, as shown in the diagram below. Thus, the unit vector of a is 1ˆ 3==aaa a . Magnitude of u, u a |a| = 3 1 a 1 1 1 a

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Notes Page 3 of 35 §2 Vectors in Two- and Three-Dimensions Key Questions: ☐ How are 2D and 3D vectors expressed algebraically? ☐ What is a position vector? ☐ How do we add two vectors? ☐ How do we subtract one vector from another vector? ☐ What is a displacement vector? ☐ When are two non-zero vectors considered to be equal to each other? ☐ How do we multiply a vector by a scalar? ☐ How do we show that two non-zero vectors are parallel? ☐ How do we show that three distinct points are collinear? ☐ What are the laws of vector algebra? ☐ How are the magnitudes of 2D and 3D vectors calculated? ☐ When and how do we apply the Ratio Theorem? 2.1 Vectors in the Cartesian Plane & Euclidean 3D Space How are 2D and 3D vectors expressed algebraically? 2D Vectors (in the Cartesian plane) 3D Vectors (in the Euclidean space) OP x y ⎯⎯ → =+ ij or x y   , OP x y z ⎯⎯ → = + +i j k or x y z    , where i and j are the unit vectors in the positive x- and y- directions respectively, i.e. where i, j and k are the unit vectors in the positive x-, y- and z- directions respectively, i.e. 1 0 =  i and 0 1 =  j 1 0 0  =  i , 0 1 0  =  j and 0 0 1  =  k The scalars x, y and z are called the Cartesian components of the vector .OP ⎯⎯ → What is a position vector? Definition 2.1 (Position Vector) With reference to a fixed point O (called the origin), the position vector of a point P relative to O is the vector .OP ⎯⎯ →

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Notes Page 4 of 35 2.2 Vector Addition How do we add two vectors geometrically? Triangle Law of Vector Addition Parallelogram Law of Vector Addition AB BC AC ⎯⎯ → ⎯⎯ → ⎯⎯ → += If PQRS is a parallelogram, PQ PS PR ⎯⎯ → ⎯⎯ → ⎯⎯ → += How do we add two vectors algebraically? Example 2.1 Given that 23OA ⎯⎯ → = + +i j k and 2,AB ⎯⎯ → =− ik find the vector .OB ⎯⎯ → 12 12 12 12 12 12 xx yy zz xx yy zz        +          + =+ + Solution ( ) 12 20 31 1 2 3 2 0 2 3 1 2 OB OA AB ⎯⎯ → ⎯⎯ → ⎯⎯ → =+        =+      −    +   =+=  +−  Example 2.2 Points O, A, B and P are such that 4 3 ,OA ⎯⎯ → = + −i j k 5OB ⎯⎯ → =−ij and OAPB is a parallelogram. Find OP ⎯⎯ → . Solution 15 41 30 6 3 3 OP OA OB ⎯⎯ → ⎯⎯ → ⎯⎯ → =+        = + −      −     = − A B C P Q R S

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Notes Page 5 of 35 2.3 Vector Subtraction How do we subtract one vector from another vector geometrically? From the diagram, AB BC AC ⎯⎯ → ⎯⎯ → ⎯⎯ → += . Therefore, BC AC AB ⎯⎯ → ⎯⎯ → ⎯⎯ → =− . How do we subtract one vector from another vector algebraically? Example 2.3 Given that 22OA ⎯⎯ → = − +i j k and 3 2 4 ,OB ⎯⎯ → = + −i j k find the vector .AB ⎯⎯ → 12 12 12 12 12 12 xx yy zz xx yy zz        −          − =− − Solution ( ) 31 22 42 31 22 42 2 4 6 OB OA AB AB OB OA ⎯⎯ → ⎯⎯ → ⎯⎯ → ⎯⎯ → ⎯⎯ → ⎯⎯ → =+ =−        = − −      −    − = − −−−  = − What is a displacement vector? See E.g. 2.3. Definition 2.2 (Displacement Vector) The displacement vector from point A to point B is the vector AB ⎯⎯ → with start point A and end point B. In general, AB OB OA ⎯⎯ → ⎯⎯ → ⎯⎯ → =− A v – u C v B

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Notes Page 6 of 35 2.4 Equality of Vectors When are two non-zero vectors considered to be Definition 2.3 (Equality of Vectors) Two non-zero vectors are said to be equal if they have the same magnitude and the same direction. equal? Example 2.4 The four distinct points A, B, C and D are such that 34AB p ⎯⎯ → = + +i j k and ( ) ( ) ( )2 2 ,DC p q q p q ⎯⎯ → = + + − + +i j k where p and q are real constants. Determine whether ABCD can be a parallelogram. 12 12 12 12 12 12 xx yy zz xx yy zz        =          = =  = Solve two of the three equations simultaneously first. Substitute the values obtained into the LHS & RHS of the third equation separately to check if it is also satisfied. Solution For ABCD to be a parallelogram, the vectors AB ⎯⎯ → and DC ⎯⎯ → must have the same direction and the same magnitude. Thus the two vectors have to be equal to each other. Hence 3 3 (1) 2 2 (2) 4 2 4 2 (3) p q p q p q p q p q p q + = +        = −  = −         + = +     (1) – (2): 32 21 1 .2 pp p p −=+ = = Substituting 1 2p = into equation (2), 15 222 qq= −  = . Substituting 1 2p = and 5 2q = into equation (3), LHS of (3) 4, 1 5 7RHS of (3) 2 LHS of (3).2 2 2 = = + =  Since there are no values of p and q that satisfy all three equations simultaneously, ABCD cannot be a parallelogram.

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