Overview

This is a problem of computing the dot product (sum of component-wise products) of two 7-dimensional vectors.

Discussion

The dot product is obtained by multiplying the corresponding components of two vectors and summing all the results.

For example, if \(\mathbf{a} = (1, 2, 3, 4, 5, 6, 7)\) and \(\mathbf{b} = (2, 0, 1, 0, 1, 0, 1)\):

\[\mathbf{a} \cdot \mathbf{b} = 1 \times 2 + 2 \times 0 + 3 \times 1 + 4 \times 0 + 5 \times 1 + 6 \times 0 + 7 \times 1 = 2 + 0 + 3 + 0 + 5 + 0 + 7 = 17\]

Since the dimension of the vectors is fixed at \(7\) (which is small) and the absolute value of each component is at most \(100\) (also small), a straightforward computation is sufficient. No special algorithms or optimization techniques are needed.

Algorithm

1. Read the first line and store the 7 integers in a list \(a\).
2. Read the second line and store the 7 integers in a list \(b\).
3. Sum up the products of corresponding components \(A_i \times B_i\) (\(i = 1, 2, \ldots, 7\)).
4. Output the total sum.

In the code, Python’s zip(a, b) is used to extract pairs of corresponding elements \((A_i, B_i)\) from the two lists, compute the product \(x \times y\) for each pair, and sum them up using sum.

a = list(map(int, input().split()))  # Read the components of vector a
b = list(map(int, input().split()))  # Read the components of vector b
print(sum(x * y for x, y in zip(a, b)))  # Compute and output the dot product

To illustrate how zip(a, b) works concretely: when \(a = [1, 2, 3, ...]\) and \(b = [4, 5, 6, ...]\), it generates the pairs \((1, 4), (2, 5), (3, 6), \ldots\) in order.

Complexity

• Time complexity: \(O(N)\) (Since \(N = 7\) is fixed, this is effectively \(O(1)\))
• Space complexity: \(O(N)\) (Used to store the two lists; since \(N = 7\), this is effectively \(O(1)\))

Implementation Notes

• Using zip allows you to pair corresponding elements of two lists and iterate over them concisely. You can achieve the same result using a for loop with indices (for i in range(7): result += a[i] * b[i]), but using zip is more Pythonic.

• Since the absolute value of each component is at most \(100\) and the dimension is \(7\), the absolute value of the dot product is at most \(100 \times 100 \times 7 = 70000\), so there is no need to worry about integer overflow.

  Source Code

a = list(map(int, input().split()))
b = list(map(int, input().split()))
print(sum(x * y for x, y in zip(a, b)))

This editorial was generated by claude4.6opus-thinking.