Geometry, Vectors and Transformations

Geometry, Vectors and Transformations

Geometry is the study of shapes of various sort. The simplest shape is the point . (It's quite difficult to explain what a point is, it is basically just a position, for instance, the very end of your nose is a point). Another simple shape is a straight line . A straight line is just the simplest shape joining two points together. A plane is a more complicated shape, it is a flat sheet, like a piece of paper or a wall. There are more complicated shapes, called solids , like a cube or a sphere. Here are some pictures of these things.

Simple geometric figures

If you have a line and a plane, you can find the point where the line cuts through the plane. In fact, sometimes you can't find the intersection, because they don't meet and sometimes the line is inside the plane so they meet at every point on the line , but this doesn't happen in the cases we're interested in. We call this the intersection of the line and the plane. Here is a picture of what this looks like.

Intersection of a line and a plane

A vector is a mathematical way of representing a point. A vector is 3 numbers, usually called x, yand z. You can think of these numbers as how far you have to go in 3 different directions to get to a point. For instance, put one arm out pointing to the right, and the other pointing straight forward. I can now give you a vector and you'll be able to find the point I'm talking about. For instance, if I say x=3, y=1, z=5, you find the point by walking 3 metres in the direction of your right hand, then 1 metre in the direction of your left hand, and then getting a ladder and climbing up 5 metres. Here is a picture of a vector.

Picture of a vector and directions

Vectors are written as (x,y,z), for instance (1,2,3) means move 1 in the x-direction, 2 in the y-direction and 3 in the z-direction.

One confusing thing about vectors is that they are sometimes used to represent a point, and sometimes they are used to represent a direction. The vector (1,0,0) can mean both ``the point you get to if you move 1 unit in the x-direction from the starting point'', or it can mean ``move 1 unit in the x-direction from where you are now''.