Force

2D Force Representation

$\overrightarrow{F_x}=\vec{F}cos(\theta )$ $\overrightarrow{F_y}=\vec{F}sin(\theta )$

3D Force Representation

Steps

1. Determine the position vector from TAIL to HEAD of the force, labeled $\overline{AB}$
2. Determine the unit vector along the position vector, by dividing each of the components of the position vector by the norm of the position vector

${\hat{e}}_{AB}=\frac{\overline{AB}}{|\overline{AB}|}$

3. Determine the force vector, by multiplying the norm (magnitude) of the force by the unit vector

$\overrightarrow{F_{AB}}=|F|{\hat{e}}_{AB}$

PDF Visual Example

Other method, Use Projections with Dot Product

I really didn’t get this, because I didn’t really understand the dot product well.

By definition, the dot product is $\vec{F}\cdot \vec{Q}=|\vec{F}||\vec{Q}|cos\theta$

This helped me wrap my head around it Project Force F in the Q direction.

$\hat{u}=\frac{\vec{Q}}{|\vec{Q}|}$

$\vec{F}\cdot \hat{u}=|\vec{F}||\hat{u}|cos\theta =|\vec{F}|cos\theta$