The Dot Product

The dot product is simply a way to multiply two vectors and get a single number that says how much they point the same way. It's called "dot" just because we write it with a dot, $\vec a\cdot\vec b$. It's good for finding angles, checking whether two directions are perpendicular, and working out how much of a force actually helps move something. The formula $\vec a\cdot\vec b = \|\vec a\|\,\|\vec b\|\cos\theta$ looks like a mouthful, but it's three simple pieces: $\|\vec a\|$ and $\|\vec b\|$ (how long each arrow is) and $\cos\theta$ (a score for how aligned they are: $1$ for the same way, $0$ for sideways, $-1$ for opposite). Multiply them and you're done; the rest of the lesson is how to use it.

We know how to add vectors and scale them by numbers. Can we multiply two vectors? There are two useful ways, and this lesson is the first one.

Two formulas, one number. The dot product has two faces. The component formula, $a_1b_1 + a_2b_2 + a_3b_3$, is how you compute it. The cosine formula, $\|\vec a\|\|\vec b\|\cos\theta$, is what it means. It's a little surprising they agree; the reason is the law of cosines applied to the triangle with sides $\vec a$, $\vec b$ and $\vec a - \vec b$. Put the two together and you can find the angle between any two vectors without drawing anything:

Orthogonal (perpendicular) vectors have $\vec a\cdot\vec b = 0$, because $\cos 90° = 0$. You'll use this test more than anything else in vector geometry: whenever a problem says "perpendicular," set a dot product to zero.

A quick sanity check on the meaning: $\vec a\cdot\vec a = \|\vec a\|^2$. A vector points exactly the same way as itself, so $\cos\theta = 1$, and you're left with its length times its length.

Projections. Think of $\|\vec b\|\cos\theta$ as the length of $\vec b$'s shadow on the line through $\vec a$. Dividing the dot product by $\|\vec a\|$ gives exactly that, and dividing by $\|\vec a\|$ once more turns it into a multiple of $\vec a$. That's the projection formula above. It splits $\vec b$ into a piece along $\vec a$ and a piece perpendicular to it.

Work. This is where the shadow idea earns its keep. A constant force $\vec F$ moving an object along $\vec d$ does work $W = \vec F\cdot\vec d$. Only the part of the force along the motion counts, and the dot product picks out exactly that part.

Lighting in 3D graphics uses $\vec n\cdot\vec \ell$ (surface normal dot light direction) to decide how bright each pixel is, and search engines compare documents by the cosine of the angle between word-count vectors.