Find the Unit Vector of a Three Dimensional Vector

Question

How can I find the unit vector of a three dimensional vector? For example, I have a problem that I am working on that tells me that I have a vector $\hat{r}$ that is a unit vector, and I am told to prove this fact:

$\hat{r} = \frac{2}{3}\hat{i} - \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k}$

I know that with a two-dimension unit vector that you can split it up into components, treat it as a right-triangle, and find the hypotenuse. Following that idea, I tried something like this, where I found the magnitude of the vectors $\hat{i}$ and $\hat{j}$, then using that vector, found the magnitude between ${\hat{v}}_{ij}$ and $\hat{k}$:

$\left|\hat{r}\right| = \sqrt{\sqrt{{\left(\frac{2}{3}\right)}^{2} + {\left(\frac{-1}{3}\right)}^{2}} + {\left(\frac{-2}{3}\right)}^{2}}$

However, this does not prove that I was working with a unit vector, as the answer did not evaluate to one. How can I find the unit vector of a three-dimensional vector?

Thank you for your time.

Answer 1

Since this is homework, we are not supposed to give you the answer. But one mistake you made is in your formula for the magnitude of $r$ - the inner square root needed to be squared. So the length of $r$ is simply the square root of the sum of the squares of the $i$, $j$ and $k$ lengths.

Good luck...

Answer 2

Although you already have an answer, I want to show you a visualization. The dark black vector is $\hat{r}$ and in green is the projection on the XY plane (ignoring the z-axis). In blue is only the z axis component vector. These form an orthogonal triangle and if you want to know the length of the hypotenuse ($\hat{r}$) you will need the length of the other two vectors. Now the length of the green vector you said you know how to get, and the length of the blue vector is trivial. If you work it out, you will arrive at the 3D formula for vector lengths.

PS. Sketches were done in GeoGebra 5.0 beta (which has some 3D capabilities now).