4.2.8 Vectors

Question 1

Define Vectors.

Answer 1

A data structure representing a quantity with both magnitude and direction. It can be represented as a list, function or geometric point.

Question 2

Vector representation with the set symbol.

Answer 2

Vectors can be represented using the symbol for the set they are drawn from, raised to a power equal to their number of components.

Question 3

How are vectors represented as lists?

Answer 3

v = (2,4)
list v = (2,4)

Question 4

How are vectors represented as 1D arrays?

Answer 4

v = (7,4,12)
array v [3]
v = (7,4,12)

Question 5

How are vectors represented as functions?

Answer 5

f = function to create the vector.
S = the complete set of values that the function can be applied to. ( the domain)
R = the potential outputs of the function. (the co-domain)

f : S -> R
e.g set S = {0,1,2,3}, co-domain R (real numbers)

Question 6

How are vectors represented as dictionaries?

Answer 6

A vector can be thought of as a function that maps from a value, representing a dimension, to the property in that dimension.

v = (2.4, 6.8, 4.5, 9.2)
0 |-> 2.4
1 |-> 6.8
2 |-> 4.5
3 |-> 9.2
dictionary v <- {0 : 2.4, 1 : 6.8, 2 : 4.5, 3 : 9.2}

Question 7

How is vector addition performed?

Answer 7

Add the components together to achieve translation.

Question 8

How is scalar vector multiplication performed?

Answer 8

Multiple the vector by the scalar, achieves scaling.

Question 9

What is the expression for the Convex combination of 2 vectors?

Answer 9

(a *u) + (b * v)
or
au + bv

Question 10

What are the conditions for convex combination of 2 vectors?

Answer 10

in au + bv:
where
- u and v are vectors
- a + b = 1
- a, b >= 0

Question 11

What is the dot product?

Answer 11

Multiply top vectors and add it with the bottom vectors multiplied.
Application of dot product:
cos q = (a . b) / (|a| * |b|)