Dot Product

Definition

$\vec{A} \cdot \vec{B} = | \vec{A} | | \vec{B} | \cos θ$ $(0 \leq θ \leq π)$

Assuming Cartesian Descriptions

$\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$
$\vec{B} = B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}$
$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$
$\cos θ = \frac{\vec{A} \cdot \vec{B}}{| \vec{A} | | \vec{B} |} = \frac{A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}}{| \vec{A} | | \vec{B} |}$

Dot Product as a Projection

SmartSelect_20230928_141341_Samsung Notes.jpg
Consider the unit vector $\hat{r} = \frac{\vec{r}}{| \vec{r} |}$
$\vec{F} \cdot \hat{r} = | \vec{F} | | \hat{r} | \cos θ$
Magnitude of vector $\vec{F}$ projected onto $\hat{r}$ : $\vec{F} \cdot \vec{r} = F_{| |} \to \vec{F_{| |}} = \vec{F} \cdot (\frac{\vec{r}}{| \vec{r} |}) \hat{r}$