Dot Product

The dot product is used to find a relationship between 2 Vectors. This products a scalar quantity $\overrightarrow{a}\cdot \overrightarrow{b}=|\overrightarrow{a}|\ast |\overrightarrow{b}|\ast \cos \theta$ $\overrightarrow{a}\cdot \overrightarrow{b}>0$ if $\theta <90$ $\overrightarrow{a}\cdot \overrightarrow{b}=0$ if $\theta =90$ $\overrightarrow{a}\cdot \overrightarrow{b}<0$ if $\theta >90$

Properties of Dot Product

Commutative Property

$\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{b}\cdot \overrightarrow{a}$

Distributive Property

$\vec{p}\cdot (\vec{q}+\vec{r})=\vec{p}\cdot \vec{q}+\vec{p}\cdot \vec{r}$ This means $(\vec{a}+\vec{b})\cdot (\vec{c}+\vec{d})=\vec{a}\cdot \vec{c}+\vec{a}\cdot \vec{c}+\vec{b}\cdot \vec{c}+\vec{b}\cdot \vec{d}$

Magnitudes Property

$\vec{a}\cdot \vec{a}=|\vec{a}{|}^2$

Associative Property with Scalar $k$

$k\vec{p}\cdot \vec{q}=k\vec{q}\cdot \vec{p}=k(\vec{p}\cdot \vec{q})$

Algebraic Dot Product

$\vec{a}=\overrightarrow{[a_1,a_2,a_3]}$ $\vec{b}=\overrightarrow{[b_1,b_2,b_3]}$ $\vec{a}\cdot \vec{b}=\overrightarrow{[a_1,a_2,a_3]}\cdot \overrightarrow{[b_1,b_2,b_3]}=a_1{b}_1+a_2{b}_2+a_3{b}_3$ Since all the components are scalars, no need to dot product, just regular multiply $a_1{b}_1$