In vector algebra, there is a quantity called a "Unit Vector" and it is defined as "A vector of length 1".

However the product of a vector multiplied by a scalar is a vector and a vector multiplied by a another vector is also a vector. An example of vector algebra is Faraday's Law which relates three vectors: The magnetic field, the motion of the conductor, and the induced voltage and current.

So if we know the result of a vector operation is another vector quantity, then a unit vector seems to be a superfluous concept. It's like having a "Unit Mass Of 1" or "Unit Weight Of 1", a "Unit Dollar of 1" that you multiply to get the actual mass, weight, or a dollar amount.

 

What if you want to know the component of a vector in a certain direction? You take the dot product of the vector and a unit vector in the direction of interest.

The reason that it is called a "unit" vector is that it is a unit of direction, just as "dollar" is a unit of money and "kg" is a unit of mass. You multiply a magnitude and a unit together to get a physical quantity, such as "5.2 kg". Similarly, a vector has three pieces, a magnitude, a scalar unit, and a direction unit. For instance, a velocity of "5 m/s x^" (the x^ represents x-hat, a unit vector in the positive x direction)

Also, the multiplication of two vectors MAY yield a vector -- or it may yield a scalar. Depends on what kind of product you use (i.e., dot product versus cross product).

 

So if we know the result of a vector operation is another vector quantity, then a unit vector seems to be a superfluous concept. It's like having a "Unit Mass Of 1" or "Unit Weight Of 1", a "Unit Dollar of 1" that you multiply to get the actual mass, weight, or a dollar amount.

Yes it is indeed like having an actual standard of mass, length etc.

But whatd'yaknow?
They are not superfluous but essential, which is why we actually have them and scientists have spent their entire careers refining these 'primary standards'

The international standard mass of 1kg is in Paris as is the original standard length of 1 metre.
These days we can use light to make the length standard, but we still need a mass standard.

This is because all units are comparisons (with a standard) so that when someone in Sydney measure 10kg she is measuring agaisnt the same standard as someone in Cairo and they will therefore get the same answer.

The unit vectors are an extension of this concept, that we employ when convenient.
a very common technique in engineering maths is to create a unit box cube called a 'control volume' to allow standardised calculations on pressure, stress, strain, flow, thermodynamic properties, area, volume...................................... the list is very long.
Then we sum the property throught the entire space we are interested in by seeing how many unit cubes fit in, or in the case of area how many unit squares fit in.
I'm sure you have done this on graph paper.

 

Be careful. Units can be measured in all sorts of ways. I'd think more along the lines of basis vectors and what kind of reference frame you're in.

 

Vectors on the real plane are a numbering system. Something like integers, rational numbers or real numbers. In this system, unit vectors play a role similar to the role 1 plays in the integers. In algebra a number X that has the property A X = X A = A for all A in a given numbering system is called a "unit" for that numbering system. Unit vectors have no connection to "units of measure".