Dot product of vectors with geometric definition


The scalar product of two vectors according to its geometric definition is the multiplication of their modules by the cosine of the angle formed by both vectors.

In other words, the dot product of two vectors is to make the product of the modules of both vectors and the cosine of the angle.

Scalar product formula

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

Scalar product in geometric definition

It is called a scalar product because the result of the module will always be a scalar, in the same way that the cosine of an angle will also be. 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 know the magnitude of each vector, the modulus is calculated.So, if we multiply the magnitude of one of the vectors (v) by the magnitude of the other vector (a) by the cosine of the angle that both form, we will know how much the two vectors measure in total.


The modulus of the vector (v) times the cosine of the angle is also known as the projection of the vector v onto the vector a.

Projection See another way to calculate the dot product of two vectors


  1. Calculate the modules of the vectors.

Given any vector of three dimensions,


The formula to calculate the modulus of a vector is:


Each subscript of the vector indicates the dimensions, in this case, the vector (a) is a three-dimensional vector because it has three coordinates.

2. Calculate the cosine of the angle.

Cosine of an angle

Example of the dot product of two vectors

Calculate the scalar product of the following three-dimensional vectors knowing that the angle they form is 45 degrees.


To calculate the scalar product we first have to calculate the modulus of the vectors:

Calculation of modules

Once we have calculated the modules of the two vectors and we know the angle, we only need to multiply them:

Calculation of the dot product

Therefore, the dot product of the previous vectors is 1.7320 units.


The following vectors would look like in a three-dimensional graph would be as follows:

Vectors in three-dimensional space

For the vector (c) we can see that the z component is zero, therefore, it will be parallel to the abscissa axis. Instead, the z component of vector (b) is positive, so we can see how it slopes upward. Both vectors are in the quadrant of the positives in terms of the component, since it is positive and is the same.

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