np.dot() for dot product in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of np.dot() with 1D arrays

What is the output of this code using np.dot() on two 1D arrays?

NumPy

import numpy as np

x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
result = np.dot(x, y)
print(result)

A TypeError

B [4 10 18]

C 32

D [24 30 36]

💡 Hint

Recall that np.dot() between two 1D arrays calculates the sum of products of corresponding elements.

The dot product of two 1D arrays is the sum of the products of their elements: (1*4) + (2*5) + (3*6) = 4 + 10 + 18 = 32.

❓ Predict Output

intermediate

np.dot() with 2D arrays (matrix multiplication)

What is the output of this code using np.dot() on two 2D arrays?

NumPy

import numpy as np

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.dot(A, B)
print(result)

A ValueError

B

[[12 16]
 [28 40]]

C

[[5 12]
 [21 32]]

D

[[19 22]
 [43 50]]

💡 Hint

Remember that np.dot() on 2D arrays performs matrix multiplication.

Matrix multiplication result is calculated as:
First row, first column: 1*5 + 2*7 = 19
First row, second column: 1*6 + 2*8 = 22
Second row, first column: 3*5 + 4*7 = 43
Second row, second column: 3*6 + 4*8 = 50

❓ data_output

advanced

Shape of np.dot() result with mixed dimensions

Given these arrays, what is the shape of the result from np.dot(A, B)?

NumPy

import numpy as np

A = np.array([[1, 2, 3], [4, 5, 6]])  # shape (2, 3)
B = np.array([7, 8, 9])                # shape (3,)
result = np.dot(A, B)
print(result.shape)

A (3,)

B (2,)

C (2, 3)

D ValueError

💡 Hint

When dot product involves a 2D array and a 1D array, the result shape depends on the 2D array's rows.

Dot product of (2,3) and (3,) arrays results in a 1D array with length 2, so shape is (2,).

🔧 Debug

advanced

Error raised by np.dot() with incompatible shapes

What error does this code raise when using np.dot() with incompatible shapes?

NumPy

import numpy as np

A = np.array([1, 2])  # shape (2,)
B = np.array([[3, 4], [5, 6], [7, 8]])  # shape (3, 2)
result = np.dot(A, B)
print(result)

A ValueError: shapes (2,) and (3,2) not aligned: 2 (dim 0) != 3 (dim 0)

B TypeError: unsupported operand type(s) for *: 'int' and 'list'

C IndexError: index out of bounds

D No error, output is [17 20]

💡 Hint

Check the dimensions of arrays to see if they can be multiplied.

np.dot() requires the inner dimensions to match. Here, (2,) and (3,2) do not align because 2 != 3.

🚀 Application

expert

Using np.dot() to compute cosine similarity

You want to compute the cosine similarity between two vectors a and b using np.dot(). Which option correctly computes it?

NumPy

import numpy as np

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

# Choose the correct formula for cosine similarity

A np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))

B np.dot(a, b) * (np.linalg.norm(a) + np.linalg.norm(b))

C np.dot(a, b) / (np.linalg.norm(a) + np.linalg.norm(b))

D np.dot(a, b) - (np.linalg.norm(a) * np.linalg.norm(b))

💡 Hint

Cosine similarity is the dot product divided by the product of vector lengths.

Cosine similarity formula: (a · b) / (||a|| * ||b||), where ||a|| is the length (norm) of vector a.

Practice

1. What does the np.dot() function do when applied to two 1D arrays (vectors)?

easy

A. Multiplies each element of the first array by the second array as a whole.

B. Adds the two arrays element-wise.

C. Returns the cross product of the two vectors.

D. Calculates the sum of products of corresponding elements (dot product).

Solution

1. Step 1: Understand np.dot() with 1D arrays

   When given two 1D arrays, np.dot() multiplies each pair of elements and sums them up.

2. Step 2: Compare with other operations

   Adding element-wise or cross product are different operations; np.dot() specifically does the sum of products.

3. Final Answer:

   Calculates the sum of products of corresponding elements (dot product). -> Option D

4. Quick Check:

   np.dot(vector1, vector2) = sum of element-wise products [OK]

Hint: Dot product sums element-wise multiplications [OK]

Common Mistakes:

• Confusing dot product with element-wise addition
• Thinking np.dot() returns cross product for 1D arrays
• Assuming np.dot() multiplies arrays element-wise without summing

2. Which of the following is the correct syntax for the np.dot() function to compute the dot product of two numpy arrays a and b?

easy

A. np.dot(a, b)

B. a.dot(b)

C. np.dot(a + b)

D. np.dot(a * b)

Solution

1. Step 1: Recall np.dot() syntax

   The function np.dot() takes two arguments: the first and second arrays to multiply.

2. Step 2: Check each option

   np.dot(a, b) correctly calls np.dot(a, b). a.dot(b) is valid but uses method syntax, not the function. Options A and C misuse the function by passing one argument or element-wise multiplication.

3. Final Answer:

   np.dot(a, b) -> Option A

4. Quick Check:

   np.dot(array1, array2) is correct syntax [OK]

Hint: Use np.dot(a, b) with two arguments [OK]

Common Mistakes:

• Passing only one argument to np.dot()
• Using addition or multiplication inside np.dot() incorrectly
• Confusing method call with function call

3. What is the output of the following code?

import numpy as np
x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
result = np.dot(x, y)
print(result)

medium

A. [4 10 18]

B. 32

C. 15

D. Error

Solution

1. Step 1: Calculate element-wise products

   Multiply corresponding elements: 1*4=4, 2*5=10, 3*6=18.

2. Step 2: Sum the products

   Sum: 4 + 10 + 18 = 32.

3. Final Answer:

   32 -> Option B

4. Quick Check:

   Sum of products = 32 [OK]

Hint: Multiply and sum elements for dot product [OK]

Common Mistakes:

• Printing element-wise multiplication instead of sum
• Confusing dot product with addition
• Expecting a vector output instead of a scalar

4. Identify the error in this code snippet:

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([5, 6, 7])
result = np.dot(A, B)
print(result)

medium

A. Syntax error in np.dot() call.

B. np.dot() requires three arguments.

C. Shape mismatch: cannot multiply 2x2 matrix with length 3 vector.

D. No error; output is [17 39].

Solution

1. Step 1: Check shapes of arrays

   Matrix A is 2x2, vector B has length 3. For dot product, inner dimensions must match.

2. Step 2: Identify mismatch

   2 (columns of A) does not equal 3 (length of B), so multiplication is invalid.

3. Final Answer:

   Shape mismatch: cannot multiply 2x2 matrix with length 3 vector. -> Option C

4. Quick Check:

   Matrix columns must match vector length [OK]

Hint: Check matrix columns match vector length [OK]

Common Mistakes:

• Ignoring shape mismatch and expecting output
• Thinking np.dot() can auto-adjust shapes
• Confusing syntax error with shape error

5. Given two matrices:

A = np.array([[1, 0, 2], [3, 1, 0]])
B = np.array([[2, 1], [0, 3], [1, 4]])

What is the result of np.dot(A, B)?

hard

A. [[4 9]
[6 6]]

B. [[2 1 8]
[6 3 0]]

C. [[2 1]
[0 3]
[1 4]]

D. Error due to shape mismatch

Solution

1. Step 1: Verify shapes for multiplication

   A is 2x3, B is 3x2, so multiplication is valid (3 matches 3).

2. Step 2: Calculate dot product manually

   Row 1 of A and column 1 of B: 1*2 + 0*0 + 2*1 = 2 + 0 + 2 = 4
   Row 1 of A and column 2 of B: 1*1 + 0*3 + 2*4 = 1 + 0 + 8 = 9
   Row 2 of A and column 1 of B: 3*2 + 1*0 + 0*1 = 6 + 0 + 0 = 6
   Row 2 of A and column 2 of B: 3*1 + 1*3 + 0*4 = 3 + 3 + 0 = 6

3. Final Answer:

   [[4 9] [6 6]] -> Option A

4. Quick Check:

   Matrix multiplication sums products of rows and columns [OK]

Hint: Multiply rows of A by columns of B and sum [OK]

Common Mistakes:

• Mixing up rows and columns during multiplication
• Expecting element-wise multiplication output
• Ignoring shape compatibility rules