Vectors Flashcards
Vector
A vector describes the movements from one point to another (a line, essentially).
Cartesian Co-ordinates
Can describe a line from the origin, called 2-vectors.
N-vector
Lets say we have a 2-vector. This can be represented as <2, 5>. This can be represented on a 2-dimension scale.
We can also have 4-vector, which is represented by <4, 5, 3, 1>.
The n-vector also exists, and is represented by some x in which x = <x1, x1, …, xn>.
This was of representing vectors is not a set, as <1, 1> are two co-ordinates and cannot be condensed down, whilst this as a set would be [1].
Origin
0 on x and y axis.
Addition
For non n vectors, we can have vector x + vector y = vector z
For n-vectors, we have w = u + v : wk = uk + vk for all of k 1 <= k <= n.
Scalar Multiplication
For non n-vectors, it is simply for some a = alpha and x = vector, a*x.
For n-vectors, we have w = av : wk = avk for all of k 1 <= k <= n (where a is a constant).
Scalar Multiplication
For non n-vectors, it is simply for some a = alpha and x = vector, a*x.
For n-vectors, we have w = av : wk = avk for all of k 1 <= k <= n (where a is a constant).
Size
For non n-vectors, it is (||u||).
For n-vectors, see obsidian.
Product
For non n-vectors, it is (u . v = ||u|| * ||v|| * cosθ)
For n-vectors, see obsidian.
Vector Attributes
Have a size
Have a direction
Do not have a position (all interpretation)
The length can be derived from a^2 + b^2 = c^2
Zero Vector
All components are 0.
Basis Vector
ek = all components are 0 except for the kth which is equal to 1.
Unit Vector
Vectors size/length are equal to one.
Orthogonal Vector
The product of u and v are 0 aka right angle.
Parallel, Right Angle and 180
For a 90 degree angle, the product is 0. For 180 degree it is -1, as well as parallel vectors.
