What is meant by resolution of vector ? Prove that a vector can be resolved along two given directions in one and only way.

Explanation

The process of splitting a vecotr into tow or more vectors such that their combined effect is same as that of givn vector is known as resolution of a vector.
Uniqueness of resultion of a vector : Let a vectr `vec R` is resolved in the directions of two coplanar and non parallel vector `vec A and vec B` as shown in

Suppose vector `vec R` is represented by `vec (OQ)` through O and Q draw lines parallel to `vec A and vec B` such that they meet at point P.
then `vec (OP) = lamda vec A and vec (PQ) = mu vec B`
Here ` lamda` and `mu` are scalars.
From triangle OPQ, we have
`vec (OQ) = vec (OP) + vec (PQ)implies vec R = lamda vec (A) + mu vec B " "...(i)`
To prove this relation: Let `vec R` can also be represented as.
`vec R= lamda vec A + mu vec B" "...(ii) `
Equating (i) and (ii) , we have
`lamda vec A + mu vec B = lamda . vec A + mu .vec B implies (lamda - lamda .) vec A = ( mu. - mu) vec B" "...(iii)`
As `vec A and vec B` are non zero vectors then it is clear from equation (iii),
either `lamda - lamda . = 0 or mu - mu = 0 implies lamda . = lamda and mu .= mu.`
Hence , there is one and only one way in which a vector `vec R` can be resolved in the directions of vectors `vec A and vec B.`