# Deep (learning) like Jacques Cousteau – Part 6 – Dot products

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(TL;DR: Start with two vectors with equal numbers of elements. Multiply

them element-wise. Sum the results. This is the dot product.)

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Hmmm…this is a tricky one!

Uhhh…did you know that Kendrick Lamar’s stage name used to be “K.Dot”?

Moi

Last

time,

we learnt **how to add vectors**. It’s time to learn about dot products!

# Today’s topic: dot products

Let’s define two vectors:

Let’s multiply these vectors element-wise. We’ll take the first

elements of our vectors and multiply them:

Let’s take the second elements and multiply them:

Now add the element-wise products:

This, my friends, is the **dot product** of our vectors.

More generally, if we have an arbitrary vector of

elements and another arbitrary vector also of

elements, then the dot product is:

The dot product is equivalent to

. Let’s come back to this next time

when we talk about **matrix multiplication**.

## What is that angular ‘E’ looking thing?

For anyone who doesn’t know how to read the dot product equation, let’s

dissect its right-hand side!

is the uppercase form of the Greek letter ‘sigma’. In this

context, means ‘sum’. So we know that we’ll need to add some

things.

We have and . In an earlier

post, we learnt that this refers to the th element of some vector. So

we can refer to the first element of our vector as

. We notice that also shares the same subscript . So

we know that whenever we refer to the second element in (i.e.

), we will be referring to the second element in (i.e.

).

We notice that is next to . So

we’re going to be multiplying elements of our vectors which occur in the

same position, .

We see that below our uppercase sigma there is a little . We also

notice that there is a little above it. These mean “Let

. Keep incrementing until you reach ”.

What is ? It’s the number of elements in our vectors!

If we expand the right-hand side, we get:

This looks somewhat similar to the equation from the example earlier:

Easy! These are the mechanics of dot products.

## What the hell does this all mean anyway?

For a deeper understanding of dot products (which is unfortunately

beyond me right at this moment!) please refer to this video:

The entire series in the playlist is so beautifully done. They are

mesmerising!

# How can we perform dot products in R?

Let’s define two vectors:

```
x <- c(1, 2, 3)
y <- c(4, 5, 6)
```

We can find the dot product of these two vectors using the `%*%`

operator:

```
x %*% y
```

```
## [,1]
## [1,] 32
```

What does R do if we simply multiply one vector by the other?

```
x * y
```

```
## [1] 4 10 18
```

This is the **element-wise product**! If the dot product is simply the sum

of the element-wise product, then `x %*% y`

is equivalent to doing this:

```
sum(x * y)
```

```
## [1] 32
```

In our previous posts, R allowed us to multiply vectors of different

lengths. Notice how R doesn’t allow us to calculate the dot product of

vectors with different lengths:

```
x <- c(1, 2)
y <- c(3, 4, 5)
x %*% y
```

This is the exception that gets raised:

Error in x %*% y : non-conformable arguments

# Conclusion

We have learnt the mechanics of calculating dot products. We can now

finally move onto matrices. Ooooooh yeeeeeah.

Justin

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