# About vector form of friction

I read a text on mechanics and in the chapter on friction, there written that the kinetic friction is in the form $$f_k = \mu_k F_N$$ where $f_k$ is the kinetic friction, $\mu_k$ is the kinetic friction coefficient, $F_N=mg$ is the normal force due to the weight of the object. In another chapter, there mentioned the vector form of the force. But it is confusing that if $f_k = \mu_k F_N$, should the vector form written as $\vec{f}_k = \mu_k\vec{F}_N$? It doesn't look right to me since $\vec{F}_N$ is downward but $\vec{f}_k$ is along horizontal. So what's the right way to write the vector for, of kinetic friction? How does people know that $f_k = \mu_k F_N$? From experiment?

I am guessing that if the object is moving at the direction $\hat{\mathcal{d}}$, so the vector form of kinetic friction should be

$$\vec{f}_k = -\mu_k(\vec{F}_n\cdot\hat{\mathcal{d}})\hat{\mathcal{d}}$$ Is that correct?

You're very close. The vector giving the friction force has magnitude $\mu_k F_N$ and is opposite the direction of travel, so it can be written as the product of $\mu_k F_N$ with a unit vector pointing in the direction opposite the direction of travel. Since the velocity $\vec v$ is in the direction of travel, the unit vector $-\vec v/v$, where $v$ is the object's speed, points opposite to the direction of travel, so the force of friction can be written as \begin{align} \vec f_k = -\mu_k F_N \frac{\vec v}{v} \end{align} The issue with your last expression is that $\vec F_N\cdot \hat d = 0$ since the normal force is perpendicular to the surface, and the direction of travel is parallel to the surface.
• A variation on joshphysic's expression which might be useful (I've no idea whether it is, because I haven't ever thought about doing problems like this in quite these terms before) is $\vec{f}_k = \mu_k \vec{F}_n\wedge \frac{\vec{F}_n \wedge \vec{v}}{F_n\,v}$. – WetSavannaAnimal Sep 16 '13 at 1:11
As we know, the direction is simply opposite the direction of motion (for kinetic friction; it is opposite the direction the object would be moving for static friction). If you are working on a free body diagram, you don't need to complicate the equation to include the direction since you draw the arrow to indicate the direction. If you have 1-D motion, your need to indicate direction is minimal; it will either be positive or negative. In general, if you need all mathematical rigor in 2- or 3-D motion, you can simply put $-\hat{d}$ on it.