Distance metrics (euclidean, cosine, manhattan) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Calculate Euclidean distance between two points

What is the output of this code that calculates the Euclidean distance between points (1, 2) and (4, 6)?

SciPy

from scipy.spatial import distance
point1 = [1, 2]
point2 = [4, 6]
result = distance.euclidean(point1, point2)
print(round(result, 2))

A 5.0

B 7.0

C 4.24

D 3.61

💡 Hint

Euclidean distance is the straight line distance between two points in space.

Euclidean distance between (1,2) and (4,6) is sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5.0.

❓ data_output

intermediate

Cosine distance between two vectors

What is the cosine distance between vectors [1, 0, -1] and [-1, 0, 1] using scipy?

SciPy

from scipy.spatial import distance
vec1 = [1, 0, -1]
vec2 = [-1, 0, 1]
result = distance.cosine(vec1, vec2)
print(round(result, 2))

A 0.0

B 1.0

C 2.0

D 0.5

💡 Hint

Cosine distance is 1 minus the cosine similarity.

The vectors are exact opposites, so cosine similarity is -1, thus cosine distance = 1 - (-1) = 2. Scipy cosine distance returns 2 for opposite vectors.

❓ Predict Output

advanced

Manhattan distance calculation

What is the Manhattan distance between points (3, 5) and (1, 1)?

SciPy

from scipy.spatial import distance
p1 = [3, 5]
p2 = [1, 1]
result = distance.cityblock(p1, p2)
print(result)

A 6

B 8

C 4

D 10

💡 Hint

Manhattan distance sums the absolute differences of coordinates.

Manhattan distance = |3-1| + |5-1| = 2 + 4 = 6.

❓ visualization

advanced

Visualize Euclidean vs Manhattan distance

Which option correctly plots the Euclidean and Manhattan distances between points (0,0) and (3,4) on a 2D grid?

SciPy

import matplotlib.pyplot as plt
import numpy as np

fig, ax = plt.subplots()
ax.plot([0,3], [0,4], label='Euclidean')
ax.plot([0,3,3], [0,0,4], label='Manhattan')
ax.scatter([0,3], [0,4], color='red')
ax.legend()
plt.show()

A A circle centered at (0,0) with radius 5

B A straight line from (0,0) to (3,0) and a diagonal from (3,0) to (3,4)

C Two diagonal lines crossing at (3,4)

D A line from (0,0) to (3,4) and an L-shaped path from (0,0) to (3,0) to (3,4)

💡 Hint

Manhattan distance looks like walking along grid streets, Euclidean is straight line.

Euclidean distance is the straight line connecting points. Manhattan distance is the sum of horizontal and vertical moves, forming an L shape.

🧠 Conceptual

expert

Understanding cosine distance range

Which statement about cosine distance computed by scipy.spatial.distance.cosine is correct?

A Cosine distance ranges from -1 to 1, where -1 means opposite and 1 means identical vectors

B Cosine distance ranges from 0 to 2, where 0 means identical and 2 means opposite vectors

C Cosine distance is always between 0 and 1, with 1 meaning identical vectors

D Cosine distance is the same as Euclidean distance normalized between 0 and 1

💡 Hint

Recall cosine distance = 1 - cosine similarity.

Cosine similarity ranges from -1 (opposite) to 1 (identical). Cosine distance = 1 - cosine similarity, so it ranges from 0 (identical) to 2 (opposite).

Practice

1. Which distance metric measures the straight-line distance between two points in space?

easy

A. Jaccard distance

B. Cosine distance

C. Manhattan distance

D. Euclidean distance

Solution

1. Step 1: Understand distance types

   Euclidean distance is the direct straight-line distance between two points.

2. Step 2: Compare with other distances

   Cosine measures angle difference, Manhattan sums absolute differences, Jaccard measures set similarity.

3. Final Answer:

   Euclidean distance -> Option D

4. Quick Check:

   Straight-line distance = Euclidean distance [OK]

Hint: Straight line means Euclidean distance [OK]

Common Mistakes:

• Confusing cosine with Euclidean
• Thinking Manhattan is straight-line
• Mixing Jaccard with numeric distances

2. Which of the following is the correct way to import the Euclidean distance function from scipy?

easy

A. from scipy.spatial.distance import euclidean

B. import scipy.distance.euclidean

C. from scipy.distance import euclidean

D. import euclidean from scipy.spatial

Solution

1. Step 1: Recall scipy import syntax

   The correct import path for distance functions is scipy.spatial.distance.

2. Step 2: Check each option

   from scipy.spatial.distance import euclidean uses correct syntax: from scipy.spatial.distance import euclidean. Others are invalid or incorrect syntax.

3. Final Answer:

   from scipy.spatial.distance import euclidean -> Option A

4. Quick Check:

   Correct import path = from scipy.spatial.distance import euclidean [OK]

Hint: Use 'from scipy.spatial.distance import' for distance functions [OK]

Common Mistakes:

• Using wrong module path
• Incorrect import syntax
• Trying to import from scipy.distance

3. What is the output of this code?

from scipy.spatial.distance import cosine
v1 = [1, 0, 1]
v2 = [0, 1, 0]
result = cosine(v1, v2)
print(round(result, 2))

medium

A. 1.0

B. 0.5

C. 0.0

D. 0.71

Solution

1. Step 1: Calculate cosine similarity basics

   Vectors v1 and v2 are orthogonal (no overlap), so cosine similarity is 0.

2. Step 2: Calculate cosine distance

   Cosine distance = 1 - cosine similarity = 1 - 0 = 1.0.

3. Final Answer:

   1.0 -> Option A

4. Quick Check:

   Orthogonal vectors cosine distance = 1.0 [OK]

Hint: Cosine distance = 1 - cosine similarity [OK]

Common Mistakes:

• Confusing cosine similarity with distance
• Rounding errors
• Assuming zero distance for orthogonal vectors

4. Find the error in this code snippet calculating Manhattan distance:

from scipy.spatial.distance import manhattan
a = [1, 2, 3]
b = [4, 5]
print(manhattan(a, b))

medium

A. Function name should be cityblock, not manhattan

B. Import statement is incorrect

C. Lists a and b must be the same length

D. Manhattan distance requires numpy arrays, not lists

Solution

1. Step 1: Check input vector lengths

   Vectors a and b have different lengths (3 and 2), which is invalid for distance calculation.

2. Step 2: Validate function and import

   Import and function name are correct; scipy.spatial.distance.manhattan is valid and accepts lists.

3. Final Answer:

   Lists a and b must be the same length -> Option C

4. Quick Check:

   Equal length vectors needed for distance [OK]

Hint: Vectors must be same length for distance functions [OK]

Common Mistakes:

• Assuming different length lists work
• Confusing function name with cityblock
• Thinking lists are invalid inputs

5. You have two points: p1 = [3, 4, 5] and p2 = [1, 1, 1]. Which distance metric will give the largest value?
Options:
A) Euclidean
B) Cosine
C) Manhattan
D) All equal

hard

A. Euclidean

B. Manhattan

C. Cosine

D. All equal

Solution

1. Step 1: Calculate Euclidean distance

   Euclidean = sqrt((3-1)^2 + (4-1)^2 + (5-1)^2) = sqrt(4 + 9 + 16) = sqrt(29) ≈ 5.39.

2. Step 2: Calculate Manhattan distance

   Manhattan = |3-1| + |4-1| + |5-1| = 2 + 3 + 4 = 9.

3. Step 3: Calculate Cosine distance

   Cosine similarity = (3*1 + 4*1 + 5*1) / (sqrt(3^2+4^2+5^2)*sqrt(1^2+1^2+1^2)) = 12 / (7.07*1.73) ≈ 0.98; Cosine distance = 1 - 0.98 = 0.02.

4. Final Answer:

   Manhattan -> Option B

5. Quick Check:

   Manhattan distance > Euclidean > Cosine distance [OK]

Hint: Manhattan sums absolute differences, usually largest [OK]

Common Mistakes:

• Assuming Euclidean is always largest
• Confusing cosine distance scale
• Ignoring absolute values in Manhattan