np.linalg.norm() for vector norms in NumPy - Mini Project: Build & Apply

Calculate Vector Norms Using np.linalg.norm()

📖 Scenario: Imagine you are working with a delivery company that tracks the movement of packages in 3D space. You want to calculate how far each package has moved from the origin point (0,0,0) using vector norms.

🎯 Goal: You will create a list of 3D vectors representing package movements, set a norm order, calculate the norms of each vector using np.linalg.norm(), and print the results.

📋 What You'll Learn

Create a list of 3D vectors using numpy arrays with exact values

Set a variable for the norm order (e.g., 2 for Euclidean norm)

Use a for loop to calculate the norm of each vector using np.linalg.norm() with the given order

Print the list of calculated norms

💡 Why This Matters

🌍 Real World

Calculating vector norms is useful in physics to find distances or magnitudes, in machine learning to measure errors or distances between points, and in computer graphics to calculate lengths and directions.

💼 Career

Data scientists and engineers often use vector norms to preprocess data, evaluate models, and analyze spatial data.

1

Create a list of 3D vectors

Create a list called vectors containing these three numpy arrays exactly: np.array([3, 4, 0]), np.array([1, 2, 2]), and np.array([0, 0, 5]).

NumPy

# Create the list of vectors
# Your code here

Hint

Use np.array() to create each vector and put them inside a list called vectors.

2

Set the norm order

Create a variable called norm_order and set it to 2 to calculate the Euclidean norm.

NumPy

import numpy as np

vectors = [np.array([3, 4, 0]), np.array([1, 2, 2]), np.array([0, 0, 5])]
# Set the norm order to 2
# Your code here

Hint

Set norm_order to 2 for the Euclidean norm.

3

Calculate norms of each vector

Create an empty list called norms. Use a for loop with variable vec to iterate over vectors. Inside the loop, calculate the norm of vec using np.linalg.norm(vec, ord=norm_order) and append the result to norms.

NumPy

import numpy as np

vectors = [np.array([3, 4, 0]), np.array([1, 2, 2]), np.array([0, 0, 5])]
norm_order = 2
# Calculate norms and store in a list
# Your code here

Hint

Use a for loop to go through each vector and calculate its norm with np.linalg.norm(). Append each norm to the norms list.

4

Print the list of norms

Write a print statement to display the norms list.

NumPy

import numpy as np

vectors = [np.array([3, 4, 0]), np.array([1, 2, 2]), np.array([0, 0, 5])]
norm_order = 2

norms = []
for vec in vectors:
    norm = np.linalg.norm(vec, ord=norm_order)
    norms.append(norm)
# Print the norms list
# Your code here

Hint

Use print(norms) to show the list of calculated norms.

Practice

1. What does np.linalg.norm() calculate by default when given a vector?

easy

A. The maximum element in the vector

B. The sum of all vector elements

C. The Euclidean length (distance) of the vector

D. The product of all vector elements

Solution

1. Step 1: Understand the default behavior of np.linalg.norm()

   By default, np.linalg.norm() calculates the Euclidean norm, which is the straight-line distance from the origin to the point represented by the vector.

2. Step 2: Compare with other options

   The sum, max, and product are different operations and not what np.linalg.norm() returns by default.

3. Final Answer:

   The Euclidean length (distance) of the vector -> Option C

4. Quick Check:

   Default norm = Euclidean length [OK]

Hint: Default norm is Euclidean distance, not sum or max [OK]

Common Mistakes:

• Confusing norm with sum of elements
• Thinking norm returns max element
• Assuming norm multiplies elements

2. Which of the following is the correct syntax to compute the 1-norm (sum of absolute values) of a vector v using np.linalg.norm()?

easy

A. np.linalg.norm(v, order=1)

B. np.linalg.norm(v, ord=1)

C. np.linalg.norm(v, norm=1)

D. np.linalg.norm(v, p=1)

Solution

1. Step 1: Recall the parameter name for norm order

   The parameter to specify the norm order in np.linalg.norm() is ord, not order, norm, or p.

2. Step 2: Check the correct syntax

   Using ord=1 correctly computes the 1-norm, which sums the absolute values of vector elements.

3. Final Answer:

   np.linalg.norm(v, ord=1) -> Option B

4. Quick Check:

   Use ord=1 for 1-norm [OK]

Hint: Use ord=1 to specify 1-norm in np.linalg.norm() [OK]

Common Mistakes:

• Using 'order' instead of 'ord'
• Using 'norm' or 'p' as parameter names
• Omitting the ord parameter for 1-norm

3. What is the output of the following code?

import numpy as np
v = np.array([3, 4])
norm_val = np.linalg.norm(v)
print(norm_val)

medium

A. 5.0

B. 7

C. 12

D. 1

Solution

1. Step 1: Calculate the Euclidean norm of vector [3, 4]

   The Euclidean norm is sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.0.

2. Step 2: Confirm the printed output

   The code prints the norm value, which is 5.0.

3. Final Answer:

   5.0 -> Option A

4. Quick Check:

   Euclidean norm of [3,4] = 5.0 [OK]

Hint: Euclidean norm of (3,4) is 5 by Pythagoras [OK]

Common Mistakes:

• Adding elements instead of squaring and summing
• Forgetting to take square root
• Confusing norm with sum or max

4. The following code throws an error. What is the mistake?

import numpy as np
v = np.array([1, -2, 3])
norm_val = np.linalg.norm(v, order=2)
print(norm_val)

medium

A. The parameter name should be 'ord' not 'order'

B. The vector contains negative values which cause error

C. np.linalg.norm() does not accept a second argument

D. The vector must be a list, not a numpy array

Solution

1. Step 1: Identify the parameter name error

   The parameter to specify norm order is ord, not order. Using order causes a TypeError.

2. Step 2: Confirm other options are incorrect

   Negative values are allowed, np.linalg.norm accepts second argument ord, and numpy arrays are valid inputs.

3. Final Answer:

   The parameter name should be 'ord' not 'order' -> Option A

4. Quick Check:

   Use ord=2, not order=2 [OK]

Hint: Use ord= for norm order, not order= [OK]

Common Mistakes:

• Using 'order' instead of 'ord'
• Thinking negative values cause error
• Believing np.linalg.norm takes only one argument

5. You have a dataset of 2D points stored as rows in a numpy array points. You want to normalize each point to have length 1 (unit vector). Which code correctly does this using np.linalg.norm()?

hard

A. normalized = points / np.linalg.norm(points, ord=1, axis=1)

B. normalized = points / np.linalg.norm(points, axis=0)

C. normalized = points / np.linalg.norm(points)

D. normalized = points / np.linalg.norm(points, axis=1, keepdims=True)

Solution

1. Step 1: Calculate norms along rows with correct shape

   Using np.linalg.norm(points, axis=1, keepdims=True) computes the Euclidean norm for each row and keeps the result as a column vector, allowing correct broadcasting for division.

2. Step 2: Normalize each point by dividing by its norm

   Dividing points by the norms with matching shape normalizes each row vector to length 1.

3. Final Answer:

   normalized = points / np.linalg.norm(points, axis=1, keepdims=True) -> Option D

4. Quick Check:

   Use keepdims=True for correct broadcasting [OK]

Hint: Use keepdims=True to keep norm shape for division [OK]

Common Mistakes:

• Using axis=0 instead of axis=1 for row-wise normalization
• Using norm of whole array instead of per row
• Using ord=1 instead of default Euclidean norm