Description of multiplication, scalar product and angles of Vectors
The multiplication of vectors has been briefly described in the section Vectors Calculation. It was shown that the result is not a vector but a real number (scalar product or dot product).
This section describes the calculation of the scalar product; and how the angle between the vectors can be calculated with the help of the scalar product.
For the two vectors \(\overrightarrow{x}=\left[\matrix{x_1\\⋮\\x_n}\right]\) and \(\overrightarrow{y}=\left[\matrix{y_1\\⋮\\y_n}\right]\)
the scalar produkt is defined as \(\overrightarrow{x}·\overrightarrow{y}= x_1·y_1 + ⋯ + x_n·y_n\)
Example
\(\overrightarrow{x}=\left[\matrix{1\\2\\3}\right]\) \(\overrightarrow{y}=\left[\matrix{4\\5\\6}\right]\) \(\overrightarrow{x}·\overrightarrow{y}= 1·4+2·5+3·6=4+10+18=32\)
From the scalar product described above and the Betrag of the vectors, the angle included by the vectors can be calculated.
The following formula is used for this \(\displaystyle cos ∡ (\overrightarrow{x},\overrightarrow{y})= \frac{\overrightarrow{x}·\overrightarrow{y}}{\left|\overrightarrow{x}\right|·\left|\overrightarrow{y}\right|}\)
The following example calculates the angle of the vectors
\(\overrightarrow{x} =\left[\matrix{3\\0}\right]\) \(\overrightarrow{y} =\left[\matrix{5\\5}\right]\)
\(\displaystyle cos ∡ (\overrightarrow{x},\overrightarrow{y})= \frac{\left[\matrix{3\\0}\right]·\left[\matrix{5\\5}\right]} {\left|\left[\matrix{3\\0}\right]\right|·\left|\left[\matrix{5\\5}\right]\right|} \) \(\displaystyle =\frac{3·5+0·5}{\sqrt{3^2+0^2}·\sqrt{5^2+5^2}}\)
\(\displaystyle = \frac{15}{3·\sqrt{2·5^2}}=\frac{15}{3·5·\sqrt{2}} = \frac{15}{15·\sqrt{2}}=\frac{1}{\sqrt{2}}\)
The result is \(∡ (\overrightarrow{x},\overrightarrow{y})= arccos\frac{1}{\sqrt{2}}=45°\)
The figure below shows the graphical representation of the vectors and the angle