Can anyone give me proof and complete explanation to the fact that
"the dot product of vector A with a unit vector B is the magnitude of A in the direction B."
Please do not get annoyed by my question.
Thank you

The dot product of vectors \vec A and \vec B is defined as the magnitude of A times the magnitude of B times the cosine of the angle between them. If \vec B is a unit vector, then its magnitude is 1, and the dot product of \vec A with the unit vector is equal to the magnitude of A times the cosine of the angle to the unit vector.

The vector \vec A can be decomposed into two orthognal vectors - call them \vec {A_1} and \vec {A_2} with angle \theta between \vec A and \vec {A_1}. The length of \vec {A_1} is therefore |A| \cos \theta , which is equal to the length of the projection of \vec A in the \vec {A_1} direction. The length of \vec A_1 is therefore same as the dot product of \vec A and the unit vector in the \vec A_1 direction.

Hope this helps!