Vectors… in… Space….!
p<>. Welcome to a mini-math
series about spatial vectors. In part 1, we'll introduce vectors,
put them in space, and see what they can do.
How
cool does that sound?
Seriously
now, vectors
If you're into math or physics, chances are you've seen vectors. There are a lot of different definitions and ways to represent vectors, but we'll go with the one that suits are purposes best:
Vectors are geometric objects that have both a size/magnitude and a direction/orientation in space. My favourite way of imagining a vector is like a giant arrow:
Vectors can be used to represent vector quantities- anything that requires both a magnitude and a direction. That's why you'll see a lot of vectors in physics, representing displacements, velocities, accelerations, forces, and other fun stuff. Learn more about physics here. But hey, we're not here to learn about physics!
What we're here to learn about is vectors in space. For the purposes of our lessons, a space will be the collection of all real vectors in a certain number of dimensions. For example, a 2-dimensional space will be your regular XY coordinate system, since any vector with a real magnitude and direction can fit into that space.
When we put vectors on a coordinate system, we gain a way to represent them. Normally, you may choose to represent vectors using their size (or length, say 5 metres) and direction (say North). Thus, we'll get something like this:
But on a coordinate plane, we can just place the tail of the vector on the origin and use regular coordinates to represent it. Instead of saying 5 m North, we'll call this vector (0,5):
A quick note is that the direction of the axes is given as (1,0) for the x-axis and (0,1) for the y-axis. This saves us the trouble of having to figure out annoying angles between the axes, and is very useful when we continue to other matters like vector products.
Note that just like no 2 sets of coordinates are equal (even if they're equidistant from the origin), no 2 vectors of different magnitude and/or direction are the same.
Now, space isn't limited to 2 dimensions. In the real world, we have 3 dimensions, so it's useful to represent vectors in 3 dimensions. That's really a lot easier than you'd think- all you have to do is add a Z-axis:
Now we also need to add a variable to our vectors to show their orientation in the third dimension. So while vectors in 2D are (a,b), in 3D they would be (a,b,c). That's not too complicated, is it?
And not surprisingly, if we wanted a 4D space for our vectors, we'll have them look like (a,b,c,d), and so on, in a 4-axes coordinate system. And you can have as many dimensions as you want! Although it's very practical, to be honest.
That's all you really need to know about how vectors fit into space, and how we can represent them using a simple coordinate system. There's a lot more we can do with vectors though.
In part 2, we'll see how we can figure out lengths, add and subtract vectors, and what unit vectors are! How fun would that be?
See you next time on
Vectors… in… Space….!
Post Comments