RI Vectors C2B Lecture Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ________________ Chapter 2B: Vectors II Page 1 of 17 Chapter 2B: Vectors II – Equations of Straight Lines SYLLABUS INCLUDE  Vector and cartesian equations of lines  Finding the foot of the perpendicular and distance from a point to a line  Finding the angle between two lines  Relationships between two lines (coplanar or skew) CONTENT 1 Equations of Straight Lines 1.1 Vector Equation of a Straight Line 1.2 Parametric Form of Equation of a Straight Line 1.3 Cartesian Equation of a Straight Line 2 Calculations Involving a Point and a Line 2.1 Determining whether a Point lies on a Line or a Line passes through a Point 2.2 Finding the Foot of Perpendicular from a Point to a Line and the Corresponding Perpendicular Distance 3 Calculations Involving a Pair of Lines 3.1 Parallel Lines, Intersecting Lines and Skew Lines 3.2 Acute Angle between Two Lines Appendix A: Guide on Solving Simultaneous Equations with/without a GC INTRODUCTION In this chapter, we shall see how the equation of a straight line can be expressed in three forms : vector, parametric and cartesian. We will then use the equation to solve problems involving distances, intersections and angles. 1 Equations of Straight Lines 1.1 Vector Equation of a Straight Line Recall that to find the equation of a straight line in the xy plane, we need a point which lies on the line and the gradient of the line. Similarly, to obtain a vector equation of a line, we need (i) the position vector of a point on the line, and (ii) a vector parallel to the line (known as the direction vector).

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ________________ Chapter 2B: Vectors II Page 2 of 17 A straight line l which passes through a fixed point A with position vector a and is parallel to a vector b has vector equation given by :,l   ra b  , where r represents the position vector of any point on the line l . Each real value of  gives the position vector of a point on the line l . To see how this formula arises, consider any point P on the line that is different from A . Since l is parallel to b , the vector AP  is also parallel to b  b  AP for some \{ 0 }  . Thus OP OA AP  ra b        for some \{ 0 }  . When point P is A , clearly OPra  (i.e. 0  ) Example 1 Relative to the origin O , the points , A B and C have position vectors given by 732aij k , 23 bi j k and 10 s t ci j k respectively, where s and t are constants. (a) Find a vector equation of the line passing through A and .B (b) Find a vector equation of the line passing through O and parallel to .a (c) The line passing through A and C has equati