Vector databases and embeddings: Worked examples

Before touching a single number, let us re-earn the three symbols we lean on the whole way down.

Definition The three symbols, re-anchored to a picture

$a = (a_{1}, a_{2}, \dots, a_{d})$ — a vector, just an ordered list of $d$ numbers. Picture it as an arrow from the origin to the point with those coordinates. ( $d$ = how many numbers = the number of axes.)
$a \cdot b = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{d} b_{d}$ — the dot product. Multiply matched slots, add. See Dot Product.
$∥ a ∥ = a_{1}^{2} + a_{2}^{2} + \dots + a_{d}^{2}$ — the norm (length of the arrow), from Pythagoras. See Vector Norms.

And the star of the show, Cosine Similarity: $sim (a, b) = \frac{a \cdot b}{∥ a ∥ ∥ b ∥} = cos θ$ where $θ$ is the angle between the two arrows. We use cosine — not raw dot product, not straight-line distance — because we want a number that answers exactly one question: "do these two arrows point the same way?" and ignores "how long are they?". Length carries no meaning here (a long article about dogs and a one-line note about dogs mean the same thing).

Figure — Vector databases and embeddings

Look at the figure: $cos θ$ slides from $+ 1$ (arrows on top of each other, angle $0°$ ), through $0$ (arrows at a right angle, $90°$ ), down to $- 1$ (arrows pointing exactly opposite, $180°$ ). That single dial is what every example below reads off.

The scenario matrix

Every situation this topic throws at you is one of these cells. The examples that follow are tagged with the cell they cover, so by the end no scenario is left unseen.

#	Case class	What's special	sim value you expect	Example
A	Generic positive	ordinary partial overlap	$0 < sim < 1$	Ex 1
B	Same direction, different length	one is a scaled copy	$sim = 1$ exactly	Ex 2
C	Orthogonal	share no components	$sim = 0$	Ex 3
D	Opposite direction	one is a negative multiple	$sim = - 1$	Ex 4
E	Negative but not opposite	some slots disagree	$- 1 < sim < 0$	Ex 5
F	Degenerate zero vector	one vector is all zeros	undefined (÷0)	Ex 6
G	Pre-normalized inputs	already unit length	dot product is sim	Ex 7
H	Euclidean vs cosine agree	unit vectors, ranking	$∥ a - b ∥^{2} = 2 - 2 cos θ$	Ex 8
I	Real retrieval, top- $k$	word problem, pick winner	rank by sim	Ex 9
J	Exam twist: cross-model	different embedders	meaningless — trap	Ex 10

The examples

Worked example Ex 1 — Cell A: generic positive similarity

$a = (1, 2, 2)$ , $b = (2, 0, 1)$ . Forecast: Guess before computing — do these look "somewhat related" or "totally unrelated"? Write down a number between 0 and 1.

Dot product: $1 \cdot 2 + 2 \cdot 0 + 2 \cdot 1 = 2 + 0 + 2 = 4$ . Why this step? The dot product is the raw "agreement" — matched slots multiplied and summed. It is the numerator of cosine.
$∥ a ∥ = 1^{2} + 2^{2} + 2^{2} = 9 = 3$ . Why this step? We must divide out $a$ 's length so only its direction survives.
$∥ b ∥ = 2^{2} + 0^{2} + 1^{2} = 5$ . Why this step? Same reason for $b$ .
$sim = \frac{4}{3 5} \approx 0.596$ . Why this step? Divide agreement by both lengths → pure direction match.

Verify: $35 \approx 6.708$ , and $4/6.708 \approx 0.596$ ✓. Since $0.596 \in [- 1, 1]$ and is positive, the arrows lean the same way but not identically — angle $arccos (0.596) \approx 53.4°$ . Sanity: a right angle would give $0$ , so being well below $90°$ matches "moderately similar."

Worked example Ex 7 — Cell G: pre-normalized vectors, dot product

is the similarity Pre-normalize $a = (3, 4)$ : divide by its length $5$ to get $\overset{a}{^} = (0.6, 0.8)$ . Pre-normalize $b = (0, 5)$ to get $\hat{b} = (0, 1)$ . (The hat $^$ means "unit length", i.e. norm exactly $1$ .) Forecast: If both arrows already have length 1, what shortcut appears?

Check lengths: $∥ \overset{a}{^} ∥ = 0.36 + 0.64 = 1 = 1$ ; $∥ \hat{b} ∥ = 0 + 1 = 1$ . Why this step? Confirm they really are unit vectors, so the denominator becomes $1 \cdot 1 = 1$ .
Dot: $0.6 \cdot 0 + 0.8 \cdot 1 = 0.8$ . Why this step? With denominator $1$ , the dot product alone equals cosine similarity.
$sim = \frac{0.8}{1} = 0.8$ .

Verify: Directly on the originals, $sim ((3, 4), (0, 5)) = \frac{20}{5 \cdot 5} = 0.8$ ✓ — identical. Why databases do this: normalize once at insert time, then every query is a single cheap dot product (no square roots per comparison). This is the engineering point in the parent note.

Worked example Ex 8 — Cell H: Euclidean and cosine ranking agree for unit vectors

Using $\overset{a}{^} = (0.6, 0.8)$ , $\hat{b} = (0, 1)$ from Ex 7. Forecast: Straight-line distance and cosine feel different. For unit vectors, do they rank the same?

Squared Euclidean distance: $∥ \overset{a}{^} - \hat{b} ∥^{2} = (0.6 - 0)^{2} + (0.8 - 1)^{2} = 0.36 + 0.04 = 0.40$ . Why this step? Euclidean distance is the ordinary ruler length between the two arrow-tips.
The identity for unit vectors: $∥ \overset{a}{^} - \hat{b} ∥^{2} = 2 - 2 cos θ$ . Why this step? Expand $(\overset{a}{^} - \hat{b}) \cdot (\overset{a}{^} - \hat{b}) = ∥ \overset{a}{^} ∥^{2} + ∥ \hat{b} ∥^{2} - 2 (\overset{a}{^} \cdot \hat{b}) = 1 + 1 - 2 cos θ$ .
Plug $cos θ = 0.8$ : $2 - 2 (0.8) = 2 - 1.6 = 0.40$ . Why this step? Matches step 1 exactly — the two views agree.

Verify: $0.40 = 0.40$ ✓. Because higher $cos θ$ ⇒ smaller distance (monotone), sorting by cosine and sorting by Euclidean give the same top- $k$ — but only after normalizing. That caveat is the whole point of the parent's Euclidean-vs-cosine mistake box.

Worked example Ex 9 — Cell I: real retrieval, pick the top-

$k$ Query $q$ = embedding of "How do I reset my password?". Candidate chunks with precomputed similarities:

C1 "Forgot password? Click 'Reset' on the login page." → sim $= 0.81$
C2 "Our office is open 9am–5pm." → sim $= 0.12$
C3 "Change your account password in Settings > Security." → sim $= 0.74$ Forecast: For top- $k$ with $k = 2$ , which two chunks get sent to the LLM?

Sort descending: $0.81 (C 1) > 0.74 (C 3) > 0.12 (C 2)$ . Why this step? Higher cosine = smaller angle = closer meaning-direction; retrieval keeps the closest arrows.
Take the top $k = 2$ : ${C 1, C 3}$ . Why this step? These two share the "password reset" direction; C2 is nearly orthogonal ( $0.12$ , angle $\approx 83°$ ).

Verify: $max = 0.81$ (C1), second $= 0.74$ (C3), and $C 2$ excluded since $0.12 < 0.74$ ✓. These two chunks become the retrieved context handed to the model — the "R" in Retrieval-Augmented Generation. See also Chunking Strategies for why C1 and C3 are separate focused passages.

Worked example Ex 10 — Cell J: the exam trap, cross-model similarity

A student embeds the documents with model-X (space $R^{768}$ ) and the query with model-Y (space $R^{768}$ ). They compute cosine similarity and get $0.02$ . They conclude "the document is irrelevant." Forecast: Is $0.02$ a trustworthy "irrelevant" verdict?

Both vectors are $768$ -dimensional, so the dot product runs without error. Why this step? The trap works precisely because nothing crashes — the shapes match.
But model-X and model-Y place meaning on different axes; slot $5$ might mean "animal" in X and "finance" in Y. Why this step? Cosine only means something when both arrows live in the same coordinate space. Across models the axes don't correspond, so the angle is noise.
Correct action: re-embed the query with model-X (the same model as the documents) and recompute.

Verify: The number $0.02$ is arithmetically valid but semantically meaningless — you cannot conclude anything. The parent's "same model for query and docs" rule is exactly this cell. (See Contrastive Learning for why each model's axes are self-consistent but model-specific.)

Summary dial

Reading the whole matrix back onto the cosine dial:

Cells A–E fill the dial from $- 1$ to $+ 1$ ; F is the one place off the dial (undefined); G–J are the same dial read for engineering and exam contexts.

Recall Which cell is each?

sim exactly $1$ , raw dot product large ::: Cell B — same direction, different length sim exactly $0$ with nonzero norms ::: Cell C — orthogonal / unrelated formula gives $0/0$ ::: Cell F — degenerate zero vector, undefined two different embedding models compared ::: Cell J — meaningless, must re-embed with one model unit vectors, distance ranks same as cosine ::: Cell H — $∥ \overset{a}{^} - \hat{b} ∥^{2} = 2 - 2 cos θ$

Connections

4.4.12 Vector databases and embeddings — the parent topic these examples drill.
Cosine Similarity · Dot Product · Vector Norms — the three tools re-earned here.
Retrieval-Augmented Generation — Ex 9 is its retrieval step in miniature.
Prompt Engineering — the retrieved chunks become prompt context.
Chunking Strategies — why chunks like C1, C3 are separate focused passages.
Contrastive Learning — why each model's axes are self-consistent but not shared (Ex 10).
HNSW and ANN Indexes · Curse of Dimensionality — how these similarities are searched at scale.