Lastnosti skalarnega produkta

Uporabimo komutativnost realnih števil:

$\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b}=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos\varphi=|\overset{\rightharpoonup}{b}||\overset{\rightharpoonup}{a}|\cos\varphi=\overset{\rightharpoonup}{b}\cdot\overset{\rightharpoonup}{a}$

Pokažimo, da je $(m\overset{\rightharpoonup}{a})\cdot\overset{\rightharpoonup}{b}=m(\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b})$.

Naj bo $m\ge 0$. Če je $\varphi$ kot med vektorjema $\overset{\rightharpoonup}{a}$ in $\overset{\rightharpoonup}{b}$, potem je kot med vektorjema $m\overset{\rightharpoonup}{a}$ in $\overset{\rightharpoonup}{b}$ prav tako $\varphi$. Tako imamo: $$(m\overset{\rightharpoonup}{a})\cdot\overset{\rightharpoonup}{b}=|m\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos\varphi=|m||\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos\varphi=$$ $$=m|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos\varphi=m(\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b})$$ Za $m < 0$  je kot med vektorjema $m\overset{\rightharpoonup}{a}$ in $\overset{\rightharpoonup}{b}$ enak $180^\circ-\varphi$. Sledi: $$(m\overset{\rightharpoonup}{a})\cdot\overset{\rightharpoonup}{b}=|m\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos(180^\circ-\varphi)=$$ $$=|m||\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|(-\cos\varphi)=-m|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|(-\cos\varphi)=$$ $$=m|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos\varphi=m(\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b})$$Podobno pokažemo, da je $\overset{\rightharpoonup}{a}\cdot (m\overset{\rightharpoonup}{b})=m(\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b})$.

$$\overset{\rightharpoonup}{a}\cdot (\overset{\rightharpoonup}{b}+\overset{\rightharpoonup}{c})=|\overset{\rightharpoonup}{a}|{\rm pr}_{\overset{\rightharpoonup}{a}}(\overset{\rightharpoonup}{b}+\overset{\rightharpoonup}{c})=$$ $$=|\overset{\rightharpoonup}{a}|({\rm pr}_{\overset{\rightharpoonup}{a}}\overset{\rightharpoonup}{b}+{\rm pr}_{\overset{\rightharpoonup}{a}}\overset{\rightharpoonup}{c})=|\overset{\rightharpoonup}{a}|{\rm pr}_{\overset{\rightharpoonup}{a}}\overset{\rightharpoonup}{b}+|\overset{\rightharpoonup}{a}|{\rm pr}_{\overset{\rightharpoonup}{a}}\overset{\rightharpoonup}{c}=$$ $$=\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b}+\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{c}$$

$\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b}=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos 90^\circ=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cdot 0=0$

Če je $\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b}=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cos \varphi=0$, potem mora biti $\cos \varphi=0$ ali pa dožina katerega izmed vektorjev enaka $0$.

Iz $\cos \varphi=0$ izhaja $\varphi=90^\circ$, kar pomeni, da sta vektorja pravokotna.

Če je $|\overset{\rightharpoonup}{a}|=0$, potem je $\overset{\rightharpoonup}{a}$ ničelni vektor, ta pa je pravokoten na poljuben vektor.

Kot med kolinearnima vektorjema je $0^\circ$ ali $180^\circ$. 

Za $\varphi=0^\circ$ je $\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b}=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cdot 1=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|$,

za $\varphi=180^\circ$ pa $\overset{\rightharpoonup}{a}\cdot\overset{\rightharpoonup}{b}=|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|\cdot (-1)=-|\overset{\rightharpoonup}{a}||\overset{\rightharpoonup}{b}|$.