Question

what is the point of having unit vectors? why can't we care just about vector itself, why to add some additional entities?

Answer 1

vectors are quantities that have magnitude as well as direction.
unit vectors characterize the 'direction' aspect of the vector.

Answer 2

for example,
consider
\(\vec{A} =<3,4>\\\vec{B} =<6,8>\\\vec{C} =<45,60>\)

Answer 3

these are three vectors that have different magnitudes, but have the same direction.
how can we express this fact as a neat mathematical expression?

Answer 4

we use unit vectors,
let\[\hat{m}=\frac{<3~,~4>}{\sqrt{3^2+4^2}}=<\frac{3}{5}~,~\frac{4}{5}>\]

Answer 5

thus using \(\hat{m}\) we can rewrite our vectors as
\[\vec{A}=5 \hat{m} \\ \vec{B}= 10 \hat{m} \\ \vec{C}=75 \hat{m} \]

Answer 6

so its not that we have put in any 'additional' entities,
its just that we have re-written it in an alternate form by separating its 'magnitude' from its 'direction'

all those three vectors have magnitude 5,10 and 75 respectively,
and point in the direction \(\hat{m}\).

Answer 7

unit vectors are useful if you want the "projection" of some vector in a particular direction u
for example
A dot u = |A| |u| cos theta
when |u|= 1, A dot u gives us the projection

also, if we are interested in the angle between two directions, it's more convenient if the directions are represented as unit vectors.
then u dot v = |u| |v| cos theta = cos theta
or
theta = acos ( u dot v)