Provided a unit vector and Force, how can I calculate it's components? Say I have a $F=kQ_{1}Q_{2}/r^{2}$ and a direction vector $(x, y, z).$ How can I find the component forces $F_{x}$, $F_{y}$, and $F_{z}$?
 A: To find the components of any vector $\bf F$ using unit vectors,
you can use the dot product between the vector and each unit vector.
So the x-component of $\bf F$ is $\bf F\cdot \hat i$ the y-component is $\bf F\cdot \hat j$ and the z-component is $\bf F\cdot \hat k$
If you have a "direction vector" $\bf u=(x,y,z)$ then its unit vector would be $$\bf \hat u=\frac{u}{\mid u\mid}$$ so that $$\bf F\cdot \hat u$$ is the projection of $\bf F$ in the direction of the unit vector $\bf \hat u$ or the component of $\bf F$ along $\bf u$.
A: You need to know the direction of the force as well as its magnitude. The force’s component along the $x$ axis is then
$F_x = |\vec F| \cos \theta$
where $\theta$ is the angle between $\vec F$ and the $x$ axis etc.
If the force is radial i.e. $\vec F = |\vec F| \vec {\hat r}$ then its components at $(x,y,z)$ are
$\displaystyle F_x = |\vec F| \frac x r$
etc.
A: It's somewhat unclear from your question, but I interpreted $F$ to be just the magnitude of the force (a scalar), and you want to construct a force of that magnitude pointing along the given direction vector.
If the direction vector is a unit vector (a vector of length 1), then all you have to do is scale (resize) it. So it's just:
$$\vec F = F \cdot (x, y, z) = (Fx, Fy, Fz)$$
If the direction vector is not a unit vector, then you have to make it into one first:
$$\text{let }\space \vec d = (x, y, z)$$
Then it's magnitude squared is
$$d^2= \vec d \vec d = x^2 + y^2 + z^2 \space\space\text{, and}\\
d = \sqrt{x^2 + y^2 + z^2}
$$
so
$$\vec F = F \cdot \frac{\vec d}{d} = (\frac{Fx}{d}, \frac{Fy}{d}, \frac{Fz}{d})$$
