RI Vectors C2A Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _____________ Chapter 2A: Vectors I Page 1 of 20 Chapter 2A: Vectors I Vector Algebra, Ratio Theorem, Scalar and Vector Products SYLLABUS INCLUDE • Addition and subtraction of vectors, multiplication of a vector by a scalar, and their geome trical 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 • Concepts of scalar product and vector product of vectors and their properties • Angle between two vectors • Geometrical meaning of ˆ⋅an and ˆ ,×an where ˆn is a unit vector CONTENT 1 Vectors in Three Dimensions (3D Vectors) 1.1 Modulus of a Vector 2 Some Fundamental Results 2.1 Parallel Vectors and its Applications 2.2 Linear Combinations of Non- Parallel Vectors 2.3 The Ratio Theorem 3 The Scalar Product (or Dot Product) 3.1 Definition of Scalar Product 3.2 Basic Properties of Scalar Product 3.3 Perpendicular Vectors 3.4 A Formula in Cartesian Form for Scalar Product 3.5 Applications of Scalar Product 4 The Vector Product (or Cross Product) 4.1 Definition of Vector Product 4.2 Properties of Vector Product 4.3 Vector Product for Vectors in Cartesian Form 4.4 Applications of Vector Product We will now move on to study vectors in three dimensions. When it comes to operations like addition, subtraction and scalar multiplication , the situation is totally analogous to what you have learn in secondary school. The only difference is the introduction of a third (z) component within the vectors. We shall start with a recap of the types of vectors.

Raffles Institution H2 Mathematics 2025 Year 5 ________________________________________________________________________________________________ _____________ Chapter 2A: Vectors I Page 2 of 20 In general, there are two types of vectors – position vectors and free vectors. A position vector is used t o denote the location of a point. It defines the position of one point relative to another point. A free vector (also known as displacement vector) is a vecto r with no associated position. OP    and OQ   are the position vectors of points P and Q with respect to the origin O . On the other hand, the vectors a and b are free vectors. In fact, a can be represented by PR    , or any directed line segment of length a and having the same direction as a . Finally, a direction vector i s a vector that describes the direction of a line segment (this will be discussed more in the next chapter) 1 V ectors in Three Dimensions (3D Vectors) To describe a vector in three dimensions, coordinates axes are chosen in the ,xy and z−directions as shown on the right. Each vector can then be represented by giving its “components” in each of the chosen dire