7. Capstone Exercise: Computing and Comparing Metrics

Exercise overview

This is the capstone exercise for Module 3. Here you're going to compute Euclidean and Manhattan distances and cosine similarity for 2D/3D vectors, compare the results, and decide which metric to use in different cases.

Part 1: Manual calculations

The data:

A = [3, 4]
B = [6, 8]
C = [0, 5]
D = [9, 1]

Task 1.1: Compute all the Euclidean distances

Compute:

dist(A, B)
dist(A, C)
dist(A, D)
dist(B, C)

See solutions

dist(A, B) = √((6-3)² + (8-4)²) = √(9+16) = 5
dist(A, C) = √((0-3)² + (5-4)²) = √(9+1) ≈ 3.16
dist(A, D) = √((9-3)² + (1-4)²) = √(36+9) ≈ 6.71
dist(B, C) = √((0-6)² + (5-8)²) = √(36+9) ≈ 6.71

Task 1.2: Compute the cosine similarities

Compute:

cos(A, B)
cos(A, C)
cos(A, D)

See solutions

cos(A, B):

A·B = 3×6 + 4×8 = 50
||A|| = 5, ||B|| = 10
cos = 50/(5×10) = 1.0

cos(A, C):

A·C = 3×0 + 4×5 = 20
||A|| = 5, ||C|| = 5
cos = 20/25 = 0.8

cos(A, D):

A·D = 3×9 + 4×1 = 31
||A|| = 5, ||D|| = √82 ≈ 9.06
cos = 31/(5×9.06) ≈ 0.68

Part 2: Interpretation

Question 2.1:

According to Euclidean, which vector is closest to A?
According to cosine, which vector is most similar to A?

See answer

Euclidean: C is closest to A (dist = 3.16)
Cosine: B is most similar to A (cos = 1.0)

Interpretation: B has the same direction as A (only a different magnitude). If direction represents "the kind of concept", B is more similar conceptually.

Part 3: Use cases

Case 1: Text embeddings

You have embeddings of 3 documents:

Doc1: [0.6, 0.8] (normalized)
Doc2: [0.6, 0.8] (normalized)
Doc3: [0.8, 0.6] (normalized)

Query: [0.7, 0.71] (normalized)

Which metric do you use? Why?

See answer

Metric: Cosine (or the dot product, which is equivalent when they're normalized)

Why:

Normalized embeddings → only direction matters
Cosine ignores magnitude → perfect for this case
Interpretable values (0.95 vs 0.7 is clear)

Case 2: GPS coordinates

You have 3 locations (latitude, longitude):

Home: [19.43, -99.13]
Work: [19.45, -99.15]
Restaurant: [19.50, -99.10]

Which metric do you use to "find the closest place to home"?

See answer

Metric: Euclidean (or Haversine for geographic precision)

Why:

Physical distance matters
Magnitude has real meaning (kilometers)
Cosine isn't appropriate ("conceptual direction" is irrelevant here)

Part 4: Reflection

Question 4.1:

Why does semantic search use cosine instead of Euclidean?

See reflection guide

Answer:

Embeddings capture direction (the kind of concept), not magnitude
- Magnitude can be noise (frequency, text length)
- Cosine ignores magnitude → it only measures direction
It's robust in high dimensions
- In 1536D, Euclidean distances are all similar
- Angles (cosine) are more discriminative
Interpretable values
- Cosine ∈ [-1, 1] → 0.95 = very similar, 0.3 = barely similar
- Euclidean ∈ [0, ∞) → hard to interpret ("is 20 a lot or a little?")

Exercise summary

What you did:

✅ You computed Euclidean and cosine by hand
✅ You compared the results (different rankings)
✅ You decided which metric to use based on the use case
✅ You reflected on why cosine is used in semantic search

The intuition, consolidated:

"Cosine measures direction (the kind of concept), Euclidean measures distance (magnitude matters). For embeddings, only direction matters → cosine is superior."

Module 3 conclusion

Congratulations on completing Module 3: Similarity and Distance. 🎉

What you achieved:

✅ You understand Euclidean distance (the straight line)
✅ You know Manhattan and other metrics
✅ You've mastered cosine similarity (the angle)
✅ You can compare metrics (trade-offs)
✅ You know why cosine is used in AI (direction > magnitude)
✅ You see how it's used in production (Pinecone, Weaviate)
✅ You applied all of it conceptually

The module's key intuition:

Cosine similarity is the standard metric in semantic search because it measures direction (the kind of concept), not magnitude (noise). In high dimensions, cosine is more robust and more discriminative than Euclidean.

Next module: Module 4: Proximity Search — kNN conceptually, indexes (HNSW, IVF) without implementing them.