I am trying to understand what ortonormal and orthogonal means, and also how to visualize them.
If I understand correctly: lets say we have three vectors $u_1,u_2,u_3$ that make up the third dimension $R^3$, we can simplify this to make calculations more easily. The simplification is finding the orthogonal basis of the newly created three dimensional space. Simply put, we take one of these vectors, lets say $u_1$, and create another vector $w_1$ that is 90 degrees in respect to $u_1$. When added to the first vector $u_1$ we get the second vector $u_2$: $\lambda u_1+\mu w_1= \gamma u_2, $ where $\lambda, \mu, \gamma \in \mathbb{R}$ in exactly one way (say $\gamma = 5$ then there is exactly one combination of $\lambda u_1 + \mu w_1 = 5u_2$.) Then we do the same for $u_2$ (now expressed in $u_1$ and $w_1$, in order to calculate $u_3$. The combination of $u_1,w_1,w_2$ the same as one vector with two orthogonal components that we now can use to express any vector in $R^3$ created by $u_1,u_2,u_3$. We get the orthonormal basis of this space instead of the orthogonal basis by dividing all the vectors by their length, e.g $\frac{u_1}{|u_1|}$.
What is the point (geometric implication) of the orthonormal basis?
Why the orthonormal basis so superior to the orthogonal basis?
When would you choose the orthonormal over the orthogonal basis? Also when would you choose to keep the original vectors (when would it be easier)?
All I can find is that using the orthogonal basis for any $R^n, n \in \mathbb{R}$ is better than what we had first because we can more easily express any vector using vectors that are connected rather than three distinct vectors combined. Or something. Which I find to not make any sense because we could perfectly well describe $R^3$ with the orthogonal vectors, right?