What is the importance of a unit vector?

Question

So obviously, we are all familiar that a unit vector is a vector that has a magnitude of 1 that lies in the same direction as our original vector. But what is the importance of it? Why do we use it? I don't really get the point of it. We can find the magnitude of any vector but what's so important about the unit vector?

I have only seen questions that say "find the unit vector of _i, _j, _k"; however, are there any other applications for unit vector? If so how would we use them?

I am currently a high school student, who is interested to find out more about this interesting concept.

Thank you in advance.

Answer 1

One reason why unit vectors are important is the following.

Suppose that we have a unit vector $\vec u(t)$ changing with a parameter $t$ (e.g. we may think $t$ as time).

Then we can prove that the derivative $\frac d{dt}\vec u(t)$ (which is a new vector that in some sense measures how $\vec u(t)$ is changing) is actually orthogonal to $\vec u(t)$. This follows immediately taking the derivative both hands of the identity $$ \vec u(t)\cdot\vec u(t)=1 $$ which describes in terms of the scalar product the fact that the length of $\vec u(t)$ is constantly $1$. In fact the product rule gives $$ 2\frac d{dt}\vec u(t)\cdot\vec u(t)=0. $$ This fact is actually of some importance when studying curves in space when we take as $\vec u(t)$ a vector tangent to the curve rescaled so to have always length $1$.

Answer 2

Let me give you an answer which would make a physicist happy. Whenever you measure a physical quantity in certain units, whether its feet, meters, seconds, etc., you can interpret this as choosing a set of unit vectors in a certain space.

Suppose you're measuring a piece of rope. Let's say I want to assign a number and call this the length of the rope. Well to do this, I need to first decide on what it means to be "1". In other words, I need to choose units, or even more down to earth, I need to choose a measuring stick. Say my rope is measured to be 3 feet. Now you can ask, if 3 this inherent, special, God-given number assigned to this rope? No, of course not, because I made a preference towards a specific measuring stick to call "1".

Despite this, we still understand that the length of the rope is a quantity that is somehow intrinsic to this rope. The way to express this is exactly to say that the number we write down changes against how the measuring stick changes. For example, I can shrink my measuring stick from feet to inches. If I measured my rope in inches, it would turn out to be 36 inches. Similarly, if I enlarged my measuring stick to yards, I would measure my rope as 1 yard. Thus, it isn't any specific number which specifies the physical quantity known as length, but rather a transformation law!

By the way, if generalizing this discussion gives the meme "a tensor is an object which transforms like a tensor".