Vectors can use simple arithmetic expressions (+, -, *, /) to perform basic operations. Let's first look at addition, then discuss a caveat of vector arithmetics.

You can add or subtract the corresponding elements of two or more vectors of the same length together.

```
> c(1,2,3) + c(99,98,97)
[1] 100 100 100
> c(1,2,3) + c(4,5,6)
[1] 5 7 9
> c(1,2,3) - c(1,1,1)
[1] 0 1 2
```

But what would happen if all the vectors weren't of the same length? Instead of erroring out, R performs recycling.

Recycling occurs when vector arithmetic is performed on multiple vectors of different sizes. R takes the shorter vector and repeats them until it becomes long enough to match the longer one.

```
> c(1,2,3,4,5,6) + c(1,3)
[1] 2 4 3 7 6 9
```

As you can see, the `c(1,3)`

vector repeated itself to form `c(1,3,1,3,1,3)`

so that it could successfully match the previous term.

If the shorter vector is not a vector of the longer one, then a warning message appears, but the operation still takes place.

```
> c(1,2,3,4,5) + c(1,3)
[1] 2 5 4 7 6
Warning message:
In c(1, 2, 3, 4, 5) + c(1, 3) :
longer object length is not a multiple of shorter object length
```

Multiplying or dividing vectors is similar to addition and subtraction in that each corresponding element matches up and a product is formed. When the sizes differ, recycling occurs.

```
> c(1,2,3) * c(0,3,6)
[1] 0 6 18
> c(1,3,5) * c(2,4)
[1] 2 12 10
```

Warning message:
In c(1, 3, 5) * c(2, 4) :
longer object length is not a multiple of shorter object length

One small detail to notice is that these common arithmetic expressions are actually functions. Thus, they can be with a similar function notation.

```
> "*"(5,6)
[1] 30
```

We can also perform the modulo operator, which outputs the remainder after division of two numbers.

```
> c(55,54,53) %% c(3)
[1] 1 0 2
```

You can also apply linear algebra on your vectors in R. To calculate the cross product, use `crossprod()`

:

```
> crossprod(1:3, 4:6)
[,1]
[1,] 32
```

You'll notice that the return type isn't a new vector, but instead a one-dimensional matrix. We'll look at matrices in the next lesson.

This book is designed to be used as the primary textbook in a college-level first course in computing. It takes a fairly traditional approach, emphasizing problem solving, design, and programming as the core skills of computer science. However, these ideas are illustrated using a non-traditional language, namely Python.

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