The operation with the vector is graphically presented. By moving the vector endpoints the numbers can be changed.

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v_{x} =

v_{y} =

w_{x} =

w_{y} =

The dot product is defined as the product of the components and the sum of these products. The scalar product is not of the order-dependent (commutative). The scalar product name comes from the fact that the result is a scalar and not a vector.

$\overrightarrow{v}\cdot \overrightarrow{w}=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ \vdots \\ {v}_{n}\end{array}\right)\cdot \left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ \vdots \\ {w}_{n}\end{array}\right)$ $={v}_{1}\cdot {w}_{1}+{v}_{2}\cdot {w}_{2}+\dots +{v}_{n}\cdot {w}_{n}$$=\sum _{i=1}^{n}({v}_{i}\cdot {w}_{i})$

The scalar product can be interpreted as the product of the geometric projection of a vector in the direction of the other vector. Physically it means that the product is formed only with the component of the vector that is effective in the direction of another vector.

The scalar product can be expressed geometrically. Here, φ the included angle of the vectors.

$\overrightarrow{v}\cdot \overrightarrow{w}=\left|\overrightarrow{v}\right|\left|\overrightarrow{w}\right|\mathrm{cos}\phi $

For the scalar product of the distributive law apply

$\overrightarrow{u}\cdot \left(\overrightarrow{v}+\overrightarrow{w}\right)=\overrightarrow{u}\cdot \overrightarrow{v}+\overrightarrow{u}\cdot \overrightarrow{w}$

and the commutative law

$\overrightarrow{v}\cdot \overrightarrow{w}=\overrightarrow{w}\cdot \overrightarrow{v}$

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Here is a list of of further useful calculators:

Vector calculation Vector addition Vector subtraction Matrix-Vector product Inner product Complex numbers graphical Addition complex numbers graphical Multiplication complex numbers graphical Division complex numbers graphical Power complex numbers graphical