Vektor Dot Produkt

This section describes how to calculate the dot product of two vectors.

In contrast to vector multiplication, the result of multiplication to the vector dot product is not a vector, but a real number (scalar product). The dot product is a mathematical combination that assigns a number (scalar) to two vectors.

For two vectors\(\overrightarrow{x}=\left[\matrix{x_1\\⋮\\x_n}\right]\) and \(\overrightarrow{y}=\left[\matrix{y_1\\⋮\\y_n}\right]\) one defines the dot product as \(\displaystyle \overrightarrow{x}·\overrightarrow{y} \)

The individual elements of the vectors are multiplied with each other and the products added. The sum of the addition is the dot product of the vectors.

Dot product \(\displaystyle= x_1·y_1 + ⋯ + x_n·y_n\)

Example

Vectors with 3 elements

\(\displaystyle\overrightarrow{x}=\left[\matrix{1\\2\\3}\right]\)     \(\displaystyle\overrightarrow{y}=\left[\matrix{4\\5\\6}\right]\)    

\(\displaystyle\overrightarrow{x}·\overrightarrow{y}= 1·4+2·5+3·6\) \(\displaystyle=4+10+18=32\)

Vectors with 2 elements

\(\displaystyle\overrightarrow{x}=\left[\matrix{1\\2}\right]\)     \(\displaystyle\overrightarrow{y}=\left[\matrix{4\\5}\right]\)    

\(\displaystyle\overrightarrow{x}·\overrightarrow{y}= 1·4+2·5\) \(\displaystyle=4+10=14\)

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