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