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...
Physics uses mathematical models to describe physical observations. You will meet this if you continue studying physics.
The mathematics is used rigorously with all its definitions. What makes it a model is a set of "assumptions", "postulates", "principles" which pick up a subset of all the space the mathematical functions can range. I will try to demonstrate with your "vectors", showing that they model observations given some assumptions.
You ask:
First, why is it that in physics, magnitude and direction of vectors are emphasized while in linear algebra components of vectors are emphasized?
Because vector algebra is used in physics to model impacts, for example, which by observation depend on magnitude and direction. In other uses of vectors the components are emphasized , but you have to continue in physics studies to encounter examples.
Also, for elementary physics (kinematics, dynamics) is R2 and R3 the vector space from which we operate and do our calculations?
We use the real numbers for end results of calculations. In special models complex numbers are important because of the simplicity of the form, but the end result of calculations has to be in real numbers.
Finally, in physics, it is emphasized that vectors are particularly useful because they are not tied to any one coordinate system. What does this mean exactly?
It means that the origin of the vectors is assumed according to the experimental set up that needs to be modeled.
In linear algebra we always write vectors emanating from the origin of the coordinate system, so it would seem as though the vectors are tied to the coordinate system.
The vectors used in physics assume as a coordinate system for the vectors the point where the vector is used to describe a force, or a velocity depending in the model. The origin could be a function of the coordinates, i.e a moving vector.
Best Answer
The basis vectors are dimensionless quantities with magnitude 1. You create dimensional vectors to represent positions, velocities, accelerations, forces, etc. by multiplying each basis vector by a dimensional scalar and then adding together. For example,
$$\mathbf{r}=(2\,\text{cm})\hat{\mathbf{e}}+(3\,\text{cm})\hat{\mathbf{a}}$$
In other words, the components of a vector carry its dimensions. That way, the same basis vectors can be used to represent all kinds of different vectorial physical quantities.