Vector similarity metrics in Prompt Engineering / GenAI - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Cosine Similarity

Which statement best describes what cosine similarity measures between two vectors?

A The angle between the two vectors, indicating how similar their directions are regardless of length.

B The difference in length between the two vectors, ignoring their direction.

C The sum of the element-wise differences between the two vectors.

D The Euclidean distance between the two vectors.

💡 Hint

Think about how cosine similarity relates to the angle formed by vectors.

Cosine similarity measures the cosine of the angle between two vectors, focusing on their direction rather than magnitude.

❓ Predict Output

intermediate

Output of Euclidean Distance Calculation

What is the output of the following Python code that calculates Euclidean distance between two vectors?

Prompt Engineering / GenAI

import numpy as np
v1 = np.array([1, 2, 3])
v2 = np.array([4, 6, 3])
distance = np.linalg.norm(v1 - v2)
print(round(distance, 2))

A 4.47

B 5.0

C 7.0

D 3.0

💡 Hint

Calculate the square root of the sum of squared differences.

The Euclidean distance is sqrt((1-4)^2 + (2-6)^2 + (3-3)^2) = sqrt(9 + 16 + 0) = sqrt(25) = 5.0. The code rounds to 2 decimals, printing 5.0.

❓ Metrics

advanced

Choosing the Best Similarity Metric for Sparse Vectors

You have very sparse high-dimensional vectors representing text documents. Which similarity metric is generally best suited to compare these vectors?

A Jaccard similarity, because it measures overlap of non-zero elements.

B Euclidean distance, because it measures absolute distance in space.

C Manhattan distance, because it sums absolute differences.

D Cosine similarity, because it focuses on the angle and ignores magnitude differences.

💡 Hint

Think about which metric handles sparse data and presence/absence well.

Jaccard similarity is effective for sparse vectors as it measures the ratio of shared non-zero elements to total unique elements, focusing on overlap.

🔧 Debug

advanced

Debugging Incorrect Cosine Similarity Calculation

What error does this code raise when calculating cosine similarity between two vectors?

Prompt Engineering / GenAI

import numpy as np
v1 = np.array([1, 0, 0])
v2 = np.array([0, 0, 0])
cos_sim = np.dot(v1, v2) / (np.linalg.norm(v1) * np.linalg.norm(v2))
print(cos_sim)

A TypeError because np.dot cannot multiply arrays.

B ZeroDivisionError because one vector has zero length.

C ValueError because vectors have different shapes.

D No error, output is nan.

💡 Hint

Check what happens when a vector has zero length in the denominator.

Since v2 is a zero vector, its norm is zero, causing a 0/0 indeterminate form. NumPy handles this by returning NaN without raising an error.

❓ Model Choice

expert

Selecting a Similarity Metric for Image Feature Vectors

You have feature vectors extracted from images using a deep neural network. These vectors are dense and normalized to unit length. Which similarity metric is most appropriate to compare these vectors for image retrieval?

A Jaccard similarity, because it measures overlap of features.

B Euclidean distance, because it measures absolute difference in features.

C Cosine similarity, because vectors are normalized and direction matters most.

D Manhattan distance, because it sums absolute differences.

💡 Hint

Consider the effect of normalization on distance metrics.

When vectors are normalized to unit length, cosine similarity and Euclidean distance are related, but cosine similarity directly measures angular similarity, which is preferred.