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-# Data Vectors
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+# Data Vectors
+
+## The Plane
+
+Pairs of numbers can be depicted as points on a plane.
+
+The plane is normally denoted by $\mathbb{R}^2$.
+
+### Details
+
+Pairs of numbers can be depicted as points on a plane.
+
+:::note Definition
+
+A **plane** is a perfectly flat surface with no thickness and no end, it can extend forever in all directions.
+It has two-dimensions, length and width.
+We need two values to find a point on the plane.
+
+:::
+
+Normally we talk about "the plane" (or "the $xy$-plane") as the collection of all pairs of numbers and denote it by
+
+$$\mathbb{R}^2 = \{ (x,y) : x,y \in \mathbb{R} \},$$
+
+giving coordinates to each point.
+The plane is also sometimes called *The Cartesian coordinate system*, named after its inventor, the French polymath René Descartes.
+
+### Examples
+
+:::info Example
+
+Plotting the point $(2,4)$ in the $xy$-plane using R:
+
+```text
+plot(2,4,xlim=c(0,6),ylim=c(0,6),xlab="x",ylab="y",cex=2)
+text(2,4,"(2,4)",pos=4,cex=2)
+```
+
+Additional points can be added using the `points()` function:
+
+```text
+points(3,5, cex = 0.5) ## a point at (3,5)
+```
+
+:::
+
+If you have two sets of coordinates on a plane you, can calculate the distance between the two points and graph the line connecting the points.
+
+:::info Example
+
+What is the distance between the two points $(3,9)$ and $(5,1)$?
+What is the distance between the 2 points (3,9) and (5,1)?
+
+We will use the Pythagorean theorem:
+
+$$d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$$
+
+We insert our values into the formula:
+
+$$d=\sqrt{(5-3)^{2}+(1-9)^{2}}$$
+
+When we combine inside the parentheses we get:
+
+$$d=\sqrt{(2)^{2}+(-8)^{2}}$$
+
+Squaring both terms:
+
+$$d=\sqrt{4+64}$$
+
+Then we take the square root:
+
+$$d=\sqrt{68}$$
+
+The result:
+
+$$d \approx 8.2462$$
+
+:::
+
+## Simple Plots in R
+
+Graphing functions in R
+
+- `plot()` - plots a scatter plot (as a line plot)
+
+- `points()` - adds points to a plot
+
+- `text()` - adds text to a plot
+
+- `lines()` - adds lines to a plot
+
+![Fig. 3](../media/2_2_Simple_plots_in_R.png)
+
+Figure: Points on a plane, drawn with R.
+
+### Examples
+
+:::info Example
+
+```text
+plot(2,3)
+```
+
+gives a single plot and
+
+```text
+plot(2,3, xlim=c(0,5), ylim=c(0,5))
+```
+
+gives a single plot but forces both axes to range from `0` to `5`.
+
+:::
+
+:::info Example
+
+The following R commands can be used to generate a plot with two points:
+
+```text
+plot(1,2,xlim=c(0,5),ylim=c(0,5),xlab="x",ylab="y")
+points(3,1)
+text(1,2,"(1,2)",pos=4, cex=2)
+text(3,1,"(3,1)",pos=4, cex=2)
+```
+
+:::
+
+:::info Example
+
+In this example, we plot three points.
+The first two arguments of the plot function.
+The third plot was added with the points are by including vectors with a length of $2$ as the $x$ and $y$ arguments of the plot function.
+The third point was added with the points function.
+The second and third points were labeled using the text function and a line was drawn between them using the lines function.
+
+```text
+plot(c(2,3),c(3,4),xlim=c(2,6),ylim=c(1,5),xlab="x",ylab="y")
+points(4,2)
+text(3,4,"(3,4)",pos=4, cex=2)
+text(4,2,"(4,2)",pos=4, cex=2)
+lines(c(3,4), c(4,2))
+```
+
+**Note**: Note that if you are unsure of what format the arguments of an R function needs to be, you can call a help file by typing `?` before the function name (e.g. `?lines`).
+
+:::
+
+## Data
+
+Data are usually a sequence of numbers, typically called a vector.
+
+### Details
+
+When we collect data these are one or more sequences of numbers, collected into data vectors.
+We commonly think of these data vectors as columns in a table.
+
+### Examples
+
+:::info Example
+
+In R, if the command
+
+```text
+x <- c(4,5,3,7)
+```
+
+is given, then `x` contains a vector of numbers.
+
+:::
+
+:::info Example
+
+Create a function in R, give it a name `Myfunction` which takes the sum of `x` and `y`:
+
+```text
+Myfunction <- function(x,y) { sum(x,y) }
+```
+
+If you input the vectors `1:3` and `4:7` into the function it will calculate the sum of `x <- (1+2+3)` and `y <- (4+5+6+7)` as follows:
+
+```text
+> Myfunction(1:3,4:7)
+[1] 28
+```
+
+:::
+
+## Indices for a Data Vector
+
+If data are in a vector `x`, then we use **indices** to refer to individual elements.
+
+### Details
+
+If `i` is an integer then $x_i$ denotes the $i^{th}$ element of $x$.
+
+**Note**: Although we do not distinguish (much) between row- and column vectors, usually a vector is thought of as a column.
+If we need to specify the type of vector, row or column, then for vector $x$, the column vector would be referred to as $x'$ and the row vector as $x^T$.
+
+(the **transpose** of the original).
+
+### Examples
+
+:::info Example
+
+If $x=(4,5,3,7)$ then $x_1=4$ and $x_4=7$
+
+:::
+
+:::info Example
+
+How to remove all indices below a certain value in R?
+
+```text
+> x <- c(1,5,8,9,4,16,12,7,11)
+
+> x
+[1]  1  5  8  9  4 16 12  7 11
+
+> y <- x[x>10]
+
+> y
+[1] 16 12 11
+```
+
+:::
+
+:::info Example
+
+Consider a function that takes to vectors
+
+$$a \in \mathbb{R}^n, b \in \mathbb{N}^m$$
+
+as arguments with:
+
+$$n \ge m$$
+
+and:
+
+$$1 \le b_1,\dots,b_m \le n.$$
+
+The function returns the sum:
+
+$$\sum_{i = 1}^m {a_b}_i$$
+
+Long version:
+
+```text
+> fn <- function(a,b) {
++ result <- sum(a[b])
++ return(result)
++ }
+```
+
+Short version:
+
+```text
+fN <- function(a,b) sum(a[b])
+```
+
+:::
+
+## Summation
+
+We use the symbol $\Sigma$ to denote sums.
+
+In R, the sum function adds numbers.
+
+### Examples
+
+:::info Example
+
+If $x=(4,5,3,7)$
+
+then
+
+$$\sum_{i=1}^{4} x_i = x_1+x_2+x_3+x_4 = 4+5+3+7 = 19$$
+
+and
+
+$$\sum_{i=2}^{4} x_i = x_2+x_3+x_4 = 5+3+7 = 15.$$
+
+In R one can give the corresponding commands:
+
+```text
+> x <- c(4,5,3,7)
+
+> x
+[1] 4 5 3 7
+
+> sum(x)
+[1] 19
+
+> sum(x[2:4])
+[1] 15
+```
+
+:::