# vector division?

I have a question. Given a vector equation such as F = ma, how can we obtain a general expression for m, the mass? If the equation was scalar, this could easily be done by dividing F by a; however, we are dealing with vectors, and, to my knowledge, a vector divided by another vector is not defined in vector algebra. Therefore, how can we obtain a general expression for m?

## 3 Answers

You could take vector norms: $$m=\frac{|F|}{|a|}$$

If the vector $\vec F$ is parallel to the vector $\vec a$ then $m$ is simply the ratio between their norms: $$m=\frac{|\vec F|}{|\vec a|}$$ If they are not parallel, then no scalar $m$ can satisfy this equation.

You can better understand this by thinking of the equation $\vec F=m\vec a$ as a system of 3 equations (if we're talking about 3 dimensional space), i.e. $$\begin{pmatrix}F_x\\F_y\\F_z\end{pmatrix}= m\begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix}$$ as a system of equations that you want to solve of the variable $m$. Since there are 3 equations and only 1 unknown, a solution is not guaranteed to exist.

• When are force and accn. Not in same direction? – Display name Sep 8 '15 at 9:35
• If they are then i think mass varies – Display name Sep 8 '15 at 9:35
• force and acceleration is always parallel. If there is an acceleration in some direction, then there exist a force in that direction. Solution always exist. If you can't find solution to that system, then your dealing with wrong force vector. – Salihcyilmaz Sep 8 '15 at 9:41
• @Salihcyilmaz True. I was answering the general question about vectors. – yohBS Sep 8 '15 at 9:42

First, as you and others stated, there is no vector division. The first thing you need to see is that:

If there exist an acceleration in some direction, then there exist a force in same direction. This is one of newtons laws.

If force and acceleration were not in same direction, then we can't find m such that satisfies the equation as yohBS stated.

If you have two vectors, then their magnitude should be equal right?

Let $ma = F_1$, then we have

$F = F_1$ which implies $|F| = |F_1|$

Then $$|F| = |ma| = m|a|$$
where you simply divide norm of $F$ to norm of $a$.

I think what you missing is that. Think about a box has mass M. This box sits on ground. Then you apply some force F. This force is not parallel to ground. Refer to figure below:

Here, you can't divide |F| to |a|. This is because acceleration is caused by net force acting on the body. Net force is tangential component of $F$ since other component is balanced by normal force from ground. Then you need to find tangential component in the direction of acceleration. Where you find m by $$|F_t| = |ma| = m|a|$$