Answers · The mathematics underneath

Why does cosine similarity ignore how long the vectors are?

In shortCosine similarity compares direction only because dividing by both vector lengths cancels their size exactly, leaving the angle between them.

Because the dot product contains the product of the two lengths as a factor, and dividing by those same lengths cancels that factor exactly. What remains depends only on direction: multiplying either vector by any positive number leaves the cosine unchanged, while the dot product itself grows or shrinks with that factor.

\cos\theta=\frac{u\cdot v}{\lVert u\rVert\,\lVert v\rVert}.

The cancellation

The dot product has two forms. Algebraically it is the sum of coordinate products. Geometrically it is $u\cdot v=\lVert u\rVert\,\lVert v\rVert\cos\theta$ , where $\theta$ is the angle between the vectors. The cosine formula is the second form rearranged: divide the dot product by the two lengths and the length factor is gone. This is exact, not an approximation. If the stored vector is doubled, its dot product with a query is doubled and its length is doubled, so the ratio has the same value. If it is halved, both numerator and denominator are halved and the ratio again does not move. The only thing that can change the answer is a change of direction.

\cos(u,cv)=\frac{c(u\cdot v)}{\lVert u\rVert\,c\lVert v\rVert}=\frac{u\cdot v}{\lVert u\rVert\,\lVert v\rVert}=\cos(u,v)\quad(c>0).

A concrete pair shows why this is not a cosmetic rescaling. Let the query be $q=(1,0)$ , and let the stored vectors be $a=(10,9)$ and $b=(2,0.1)$ . The dot product with $q$ is the first coordinate in each case, so $a$ scores $10$ and $b$ scores $2$ . The lengths are not equal, and that is the whole story.

FIG 1

vector	dot with q	length	cosine
a=(10,9)	10	sqrt181≈13.454	10/sqrt181≈0.743
b=(2,0.1)	2	sqrt4.01≈2.0025	2/sqrt4.01≈0.999

Under the dot product $a$ wins, by $10$ against $2$ . Under the cosine $b$ wins, because $10/\sqrt{181}\approx0.743$ is smaller than $2/\sqrt{4.01}\approx0.999$ . The long vector $a$ is less aligned with the query, while $b$ is almost exactly on it. The dot product rewarded $a$ for being long enough to accumulate a large sum; the cosine asked only how well the directions line up. That is what ignoring length means in practice: the comparison is between directions, not between magnitudes.

What goes wrong

The natural guess is that dividing by the lengths only rescales the score, so the ranking should stay the same. That would be true if every pair were divided by one common number. It is not: the divisor is $\lVert u\rVert\lVert v\rVert$ , which depends on the pair. In the example the divisor for $a$ is $\sqrt{181}\approx13.454$ and for $b$ is $\sqrt{4.01}\approx2.0025$ , so the two dot products are shrunk by different amounts. A single global scaling would leave $10>2$ intact. Dividing by different numbers turns it into $0.743<0.999$ . The division is therefore doing more than changing the units of the score: it removes the length advantage that the dot product was giving to $a$ .

The cosine always lies between minus one and one. A value near one means the directions coincide, a value near minus one means they oppose, and a value near zero means perpendicular in the geometric sense. In real learned collections the scores often occupy only part of that range, because vectors cluster in a narrow cone rather than spreading over all directions. A high score can be excellent in one collection and ordinary in another, so the absolute number is less reliable than the ranking within the same collection.

When length matters

Dividing out length is a decision, not a cleanup. If length encodes how strong a feature is, how confident a model is, or how much text a document contains, the cosine throws that information away. A faint vector pointing in the right direction and an overwhelming vector pointing the same way receive the same score, so an emphatic match and a tentative one become indistinguishable. That is exactly what is wanted when two documents of different lengths discuss the same subject. It is exactly wrong when the task is to prefer the stronger signal. If every vector is first given unit length, the denominator is one and the dot product and cosine rank pairs identically; without that normalisation the choice between them is a choice about whether length should count.

Worth keeping

Scaling a vector by a positive number changes its dot product with another vector by the same factor and changes its length by that factor, so the cosine is unchanged.
Dividing by the two lengths is not a single rescaling of every score: the divisor is different for each pair, which is why it can change the ranking.
Cosine similarity throws away length on purpose, so it is the wrong choice when length encodes confidence, strength, or document size.

Where this goes next

This question is one step of Vectors, Distance and Similarity, a course that works through its subject from the beginning. The page that teaches this part of it is Comparing Direction Alone, and it reads the whole way through whether or not you have an account.

What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.

Read that page The whole course