# Provided a unit vector and Force, how can I calculate it's components? [closed]

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 $$F_{z}$$?

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 $$\bf u$$.

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 $$\vec F = F \cdot \frac{\vec d}{d} = (\frac{Fx}{d}, \frac{Fy}{d}, \frac{Fz}{d})$$

• Your equations seem to imply that you get a vector from a dot product. Sep 21 '21 at 3:56
• @josephh - no, the only dot product here is $\vec d \vec d$, producing a scalar. I'm using the OP's notation where $\vec F$ (with an arrow) is a vector, and $F$ is its magnitude. Perhaps the multiplication dot is confusing in the final row? Or my use of (x, y, z) shorthand? Sep 21 '21 at 5:13
• Yeah. I think the dot caught me off guard - then I read it again and looks OK. Cheers. Sep 21 '21 at 5:27
• Thank you, this helped a lot! Sep 22 '21 at 6:14

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.