How Much Together: Dot Product

Lesson 6 of 12 · 7 min · Free, no account needed

What you will learn

The dot product of a and b is a₁b₁ + a₂b₂: multiply in pairs, then add.
It is big when the arrows point the same way, 0 at a right angle, negative when they point apart.
It grows with b's shadow on a. Apps and neurons use it to score how well things match.

A movie app stores your taste as an arrow. How does it check if a film points your way?
Ada's taste is the blue arrow. Tap the film you think the app suggests.
Swing arrow b around a. Watch the number and the shadow.
The number is 2 × b's first part + 1 × b's second part. Reach each goal.
Sort by eye: same way, right angle, or against?
Then the number checks you.
Positive means less than a right angle, zero means exactly a right angle, negative means more.
This number is the dot product.
The dot product a · b = a₁b₁ + a₂b₂. Here 2 × 1 + 1 × 3 = 5.
a · b = 0 means the arrows are perpendicular: a right angle.
The app scores each film as taste · film and suggests the biggest. For Ada that is Laugh Riot.
Now set your own taste.
Across is comedy, up is action.
Game: movie night. Rank your top three films.
Tap the highest score first. Work it out from the arrows: the score appears after you tap.
A neuron in an AI does the same sum: weights · inputs.
The inputs are 2 and 1. Set the two weights so the sum is exactly 7.
What is (3, 2) · (4, -1)?
a · b = a₁b₁ + a₂b₂ = 3 × 4 + 2 × (-1) = 10.
Which pair is at an exact right angle?
Only (2, 1) and (1, -2) give a dot product of 0.
The dot product says how much two arrows point together.
Compare your taste with three friends. Who should you watch films with?

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