2. 2022 Vectors (Lines) Lecture Notes (For Upload)

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: VECTORS Page | 47 H2 MATHEMATICS (9758) TOPIC VECTORS (LINES) 2022/JC1 9 Introduction We’ve all marveled at the incredibly life-like computer generated images in the movies. What most of us don’t realise is that the dinosaurs of Jurassic Park and the wonders of Lord of the Rings – particularly the star turn of Gollum – wouldn’t have been possible without vector geometry and math. But how are these amazing images made? Computer graphics and computer vision are huge subjects. The first step in creating a computer generated movie is to create the characters in the story and the world they live in. Each of these objects is modelled as a surface, made up of connected flat polygons that are usually triangles, with the vertices of each triangle stored in computer memory. Now that the surface of our object is a wire mesh of triangles, we are ready to colour each of its components. Here it’s important to realistically capture the lighting of the scene we’re modelling, and this is done using a process called ray tracing. Starting from our viewpoint, we trace rays backwards towards the object and let them reflect off it. If the ray from our eye reflects off the facet (one of our wire mesh triangles) and intersects a light source, we shade that facet in a bright colour so that it appears lit up the light source. If the reflected ray does not meet the light source, we shade the facet in a darker colour. Trace a ray from your viewpoint to a facet. Does it reflect off and intersect a light source ? To trace a ray back to a particular facet, we need to describe the surface mathematically, and solve geometric equations involving the straight lines described by the ray and the plane described by that facet. This is done using vectors. In the next two chapters, we would be introduced to the mathematical representation of lines and planes. Source: https://plus.maths.org/content/os/issue42/features/lasenby/index Content Outline: Three-Dimensional Vector Geometry • 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) First objects are modelled as wire skeletons made up from simple polygons such as triangles.

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: VECTORS Page | 48 10 Equation of a Line 10.1 Vector Equation of a Line Exploration Activity Let l be the line passing through two points, P and Q , as shown in the diagram below: Suppose the position vector of P is p and the vector PQ = m   . (a) Express the position vector of Q in terms of p and m . pmOQ = +    (b) It is given that the point E lies on the line PQ produced such that 3PE P