Vector dot product

Learn it

Words to know

vector

A list of numbers, such as [1, 2, 3]. Each position is called a dimension.

dot product

One number made from two vectors: multiply the numbers at the same position, then add the products.

How it works

1. Line the two vectors up: position 0 with position 0, position 1 with position 1, and so on.
2. Multiply each pair: a[i] * b[i].
3. Add all the products. With no positions at all, the sum is 0.

Worked example

Input
a = [1, 2, 3]
b = [4, 5, 6]

Output
32

The answer is one number, 32, however long the vectors are. The biggest product comes from position 2, where both vectors are large.

Why it matters

The next lesson divides a dot product by two lengths to get cosine similarity, which is how a search system decides which document is closest to a question.

Your turn

Return the dot product of two vectors with the same number of positions.

a

a list of numbers, possibly empty; values can be negative

b

a list with as many numbers as a

Returns

One number: the sum of a[i] * b[i] over every position. For two empty lists, 0.

Input
a = [1, 2, 3]
b = [4, 5, 6]

Output
32

Input
a = [-2, 3]
b = [4, -1]

Output
-11

Input
a = []
b = []

Output
0

The checks also try

• a vector of zeros
• decimal values
• one position
• the same number twice in one vector

Hints

Each hint gives away a little more. Open one only when you need it.

1. Hint 1: Where to start

   You need one running total that starts at 0, and one loop over the positions.

2. Hint 2: The key step

   On each pass, multiply the two numbers at the same position, a[i] and b[i], and add the product to the total.

3. Hint 3: The outline

   Set total to 0. For each position i, from 0 to the last one, add a[i] * b[i] to total. After the loop, return total.

Solution

From here on the page gives the method away. If you are still solving, go back to the hints.

Step by step

1. Start the total at zero. Two empty vectors never enter the loop, so the starting value is also the answer for that case.
2. Add the product of the two numbers at each position. Each position contributes one product. A position where both numbers are large raises the total the most.
3. The total is the dot product. After the last position nothing is left to add, so the total is returned as it is.

The same numbered steps are the comments in the solution.

function solve(a, b) {
  // 1. Start the total at zero
  let total = 0;
  for (let i = 0; i < a.length; i++) {
    // 2. Add the product of the two numbers at each position
    total += a[i] * b[i];
  }
  // 3. The total is the dot product
  return total;
}

def solve(a, b):
    # 1. Start the total at zero
    total = 0
    for i in range(len(a)):
        # 2. Add the product of the two numbers at each position
        total += a[i] * b[i]
    # 3. The total is the dot product
    return total

Watch out

• Do not add the two numbers at a position. The dot product multiplies each pair first, and only then adds.
• Do not start the total at 1, as you would for a running product. Two empty vectors must give 0.

Check yourself