Question

what does the dot product allow you to solve? In other words why do we need the dot product (inner scalar product)

Answer 1

Depends on where you want to apply. It is very important in vector operations where you can project the vector to the respective axes by using the dot product. It is also used in other concepts such as the difference between covariant and contravariant vectors.

Answer 2

in case of work done when we multiple two vectors we get scalar
that's why we invented dot product
is it clear now?