Product of two vectors

economic-dictionary

The dot product of two vectors in coordinates is the sum of the product of the coordinates of each vector preserving the order of the dimensions.

In other words, the dot product in coordinates of two vectors is the result of multiplying the coordinates of the same dimension of the vectors and adding them.

It is called a dot product because the result of the multiplication will always be a scalar. The result of this multiplication will be a number that expresses a magnitude and has no direction. In other words, the result of the dot product will be a number, not a vector. Therefore, we will express the resulting number as any number and not as a vector.

To express the product of vectors in coordinates, the canonical reference system is used.

In this article we will see, all that is said, two ways to calculate the dot product of two vectors. The first has been described above, while the second we will see later.

Formula of the product of two vectors

Given two vectors:

Cartoon vector

The dot product is calculated as follows:

Scalar product

The dot product of two vectors is obtained by multiplying the coordinates of the vectors, always keeping the dimensions. In other words, you can only multiply the coordinates of the same dimension.

Correction

In the first example it is fine because we are multiplying the first coordinate of vector a and vector b. The second example is wrong because we are multiplying the first coordinate of vector a and the second coordinate of vector b. Multiplying coordinates of different dimensions is not correct.

Scalar product formula for k vectors

Given k vectors with n coordinates:

Cartoon vector

The dot product is calculated as follows:

Scalar product

Although we have many vectors with many dimensions, the dot product works in the same way: make the sum of the multiplication of the coordinates that are of the same dimension.

Steps to follow to calculate the dot product of two vectors

  1. Identify the vectors that we want to multiply and their coordinates.
  2. Multiply the coordinates of the same dimension.
  3. Add the previous multiplications.
  4. Check that the result is a single number.

Geometric definition dot product

The dot product of two vectors can also be expressed as the product of the modules of both vectors and the cosine of the angle of the vectors.

Given two vectors, the dot product is calculated as follows:

Geometric definition dot product

To delve more into this other form of calculation, we recommend that you visit the following article:

See another way to calculate the dot product of two vectors

Scalar Product Example

Calculate the dot product of the following vectors:

Example

The result of a dot product will always be a scalar, that is, a number. The result of our example matches the theory and is therefore correct.

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