# 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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