Dot Product

The dot product is a scalar, calculated as the projection of one vector onto another. An example would be to calculate the forward force on a cart, which is pulled via a rope in a forwards direction, but under an angle of $$ \alpha $$ upwards. Instead of a dot product, it is also possible to use a cosine to calculate the projection of one vector onto another, however, in multi dimensial spaces, the calculation of this angle $$ \alpha $$ can be much more work than calculating the dot product.

Calculation
The dot product $$ A \cdot B $$ can be calculated as following. Please note $$ A $$ and $$ B $$ are n-dimensional vectors.

$$ A \cdot B =

\begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_n \end{bmatrix} \cdot

\begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ b_n \end{bmatrix} =

a_1*b_1+a_2*b_2+a_3*c_3+a_n*b_n

$$

The interpretation of this scalar is: the component of vector $$ A $$ in the direction of vector $$ B $$.