Force

3D Description of Forces

$F_x=F\cos {\theta }_x$
$F_y=F\cos {\theta }_y$
$F_z=F\cos {\theta }_z$
$F=\sqrt{F_x^2+F_y^2+F_z^2}$
$\overrightarrow{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$
$=F\cos {\theta }_x\hat{i}+F\cos {\theta }_y\hat{j}+F\cos {\theta }_z\hat{k}$
$=F(\cos {\theta }_x\hat{i}+\cos {\theta }_y\hat{j}+\cos {\theta }_z\hat{k})$
$=F\hat{\lambda }$ where $\hat{\lambda }$ is a unit vector
$\left|\hat{\lambda }\right|=1$

$\overrightarrow{F}=F\hat{\lambda }=F\left(\frac{\overrightarrow{MN}}{\left|\overrightarrow{MN}\right|}\right)$
$=F\left(\frac{dx\hat{i}+dy\hat{j}+dz\hat{k}}{\sqrt{d{x}^2+d{y}^2+d{z}^2}}\right)$
Note: $d=\sqrt{d{x}^2+d{y}^2+d{z}^2}$
$=F[\left(\frac{dx}{d}\right)\hat{i}+\left(\frac{dy}{d}\right)\hat{j}+\left(\frac{dz}{d}\right)\hat{k}]$
Where $\left(\frac{dx}{d}\right)\hat{i}=\cos {\theta }_x$, $\left(\frac{dy}{d}\right)\hat{j}=\cos {\theta }_y$, $\left(\frac{dz}{d}\right)\hat{k}=\cos {\theta }_z$