I see the dot product as directional multiplication. But multiplication goes beyond repeated counting: it’s applying the essence of one item to another.

Normal multiplication combines growth rates: “3 x 4” can mean “Take your 3x growth and make it 4x larger (i.e., 12x)”. Complex multiplication lets us combine rotations. Integrals let us do piece-by-piece multiplication.

A vector is “growth in a direction”. The dot product lets us apply the directional growth of one vector to another: the result is how much we went along the original path (positive progress, negative, or zero).

Today let’s build our intuition for how the dot product works.

## Getting the Formula Out of the Way

You’ve seen the dot product equation everywhere:

And also the justification: “Well Billy, the Law of Cosines (you remember that, don’t you?) says the following calculations are the same, so they are.” Not good enough — it doesn’t click! Beyond the computation, what does it mean?

The goal is to apply one vector to another. Each computation examines this from a rectangular perspective (x- and y-coordinates) or a polar one (magnitudes and angles). The “blah = foo” equation above really means “Here’s two equivalent ways to ‘directionally multiply’ vectors”.

(Similarly, we can show that Euler’s formula (e^{ix} = cos(x) + i · sin(x)) is true because the Taylor series is the same on both sides. Accurate but unsatisfying! Instead, see how both sides can describe the same motion.)

## Seeing Numbers as vectors

Let’s start simple, and see 3 x 4 as a dot product:

The number 3 is “directional growth” in a single dimension (x-axis, let’s say), and 4 is “directional growth” in that same direction. 3 x 4 = 12 means 12x growth in that single dimension. Ok.

Now, suppose each number refers to a different dimension. Let’s say 3 means “triple your bananas” (sigh… or “x-axis”) and 4 means “quadruple your oranges” (y-axis). They’re not the same type of number: what happens when we apply growth, aka use the dot product, in our “bananas, oranges” universe?

- (3,0) is “Triple your bananas, destroy your oranges”
- (0,4) is “Destroy your bananas, quadruple your oranges”

Applying (0,4) to (3,0) means “Destroy your banana growth, quadruple your orange growth”. But (3, 0) had no orange growth to begin with, so the net result is 0 (“Destroy all your fruit, buddy”).

See how we’re “applying” and not adding. With addition, we sort of smush the items together: (3,0) + (0, 4) = (3, 4) [a vector which triples your oranges *and* quadruples your bananas].

“Application” is different. We’re mutating the original vector according to the rules in the second. And the rules are “Destroy your banana growth *rate*, and triple your orange growth *rate*“. And, sadly, this leaves us with nothing.

The final result of this process can be:

- zero: we don’t have any growth in the original direction
- positive number: we have some growth in the original direction
- negative number: we have negative (reverse) growth in the original direction

## Understanding the Calculation

“Applying vectors” is still a bit abstract. I think “How much energy/push is one vector giving to the other?”. Here’s how I visualize it:

**Rectangular Coordinates: Component-by-component overlap**

Like multiplying complex numbers, see how each x- and y-component interacts:

We list out all four combinations (x&x, y&x, x&y, y&y). Since the x- and y-coordinates don’t affect each other (like holding a bucket sideways under a waterfall — nothing falls in), the total energy absorbtion is absorbtion(x) + absorbtion(y):

**Polar coordinates: Projection**

The word “projection” is so sterile: I prefer “along the path”. How much energy is actually going in our original direction?

Here’s one way to see it:

Take two vectors, a and b. Rotate our coordinates so b is horizontal: it becomes (|b|, 0), and everything is on this new x-axis. What’s the dot product now? (It shouldn’t change just because we tilted our head).

Well, vector a has new coordinates (a1, a2), and we get:

a1 is really “What is the x-coordinate of a, assuming b is the x-axis?”. That is |a|cos(θ), aka the “projection”:

## Analogies for the Dot Product

The common interpretation is “geometric projection”, but it’s so bland. Here’s some analogies that click for me:

**Energy Absorbtion**

One vector are solar rays, the other is where the solar panel is pointing (yes, yes, the normal vector). Larger numbers mean stronger rays or a larger panel. How much energy is absorbed?

- Energy = Overlap in direction * Strength of rays * Size of panel

If you hold your panel sideways to the sun, no rays hit (cos(θ) = 0).

But… but… solar rays are leaving the sun, and the panel is facing the sun, and the dot product is negative when vectors are opposed! Take a deep breath, and remember the goal is to embrace the analogy (besides, physicists lose track of negative signs all the time).

**Mario-Kart Speed Boost**

In Mario Kart, there are “boost pads” on the ground that increase your speed (Never played? I’m sorry.)

Imagine the red vector is your speed (x and y direction), and the blue vector is the orientation of the boost pad (x and y direction). Larger numbers are more power.

How much boost will you get? For the analogy, imagine the pad multiplies your speed:

- If you come in going 0, you’ll get nothing [if you are just dropped onto the pad, there’s no boost]
- If you cross the pad perpendicularly, you’ll get 0 [just like the banana obliteration, it will give you 0x boost in the perpendicular direction]

But, if we have some overlap, our x-speed will get an x-boost, and our y-speed gets a y-boost:

Neat, eh? Another way to see it: your incoming speed is |a|, and the max boost is |b|. The amount of boost you actually get (for being lined up with it) is cos(θ), for the total |a||b|cos(θ).

**Physics Physics Physics**

The dot product appears all over physics: some field (electric, gravitational) is pulling on some particle. We’d love to multiply, and we could if everything were lined up. But that’s never the case, so we take the dot product to account for potential differences in direction.

It’s all a useful generalization: Integrals are “multiplication, taking changes into account” and the dot product is “multiplication, taking direction into account”.

And what if your direction is changing? Why, take the integral of the dot product, of course!

## Onward and Upward

Don’t settle for “Dot product is the geometric projection, justified by the law of cosines”. Find the analogies that click for you! Happy math.

## Other Posts In This Series

- Vector Calculus: Understanding Flux
- Vector Calculus: Understanding Divergence
- Vector Calculus: Understanding Circulation and Curl
- Vector Calculus: Understanding the Gradient
- Understanding Pythagorean Distance and the Gradient
- Vector Calculus: Understanding the Dot Product
- Vector Calculus: Understanding the Cross Product