what will be the result if we will find dot product of a given displacement vector to the unit vector of another displacement vector? Are we going to get the magnitude of first vector in the direction of second vector? If yes,then how and on basis of which formula & if no then what will be the outcome?
Eg. Let vector A=2i+3j
vector B=i+j unit vector=(I+j)/√2
now the result of (2i+3j).{(I+j)/2^(1/2)} is 5/√2 if this is the magnitude of vector A along the direction of vector B then what is the formula which has been used here?
thank you

the dot product id merely the product of the coefficients..
e.g.. A=5i + 3j -7k or <5,3,-7>
B= -2i + 5j -8k or <-2,5,-8>
so the dot product will be A.B.= [5*(-2)] + (3*5) + [(-7) * (-8)]= -10 + 15 + 56= 61

similalry,as per your problem
A= 2i + 3j
U= I/2^1/2 + j/2^1/2

therefore A.U. = 2*(1/2^1/2) + 3*(1/2^1/2)=2/2^1/2 + 3/2^1/2=5/2^1/2.. thats your result..
as dot product is a scalar quantity,we can not determine the direction.For that purpose we will have to calculate the vector product.

Yes - the dot product of vector A with a unit vector B is the magnitude of A in the direction B. As already noted it is a scalar, and so itself has no direction.

An alternative way to calculate the dot product is \vec A \cdot \vec B = |A||B| \cos\theta where \theta is the angle between vectors A and B. So if |B| = 1 (as for a unit vector) the dot product is magnitude of A times the cosine of the angle between A and B. This is the same as the projection of A onto an axis aligned with B.