Question

Find coordinate vector relative to basis

Answer 1

    Finding the coordinate vector of a given point (relative to) with respect to a given basis.
 
Basis vectors:

   In 3-d coordinate system we have the default basis vector system of x, y and z axes. Three vectors which are independent of each other forms a basis.       

  We can represent any vector in 3 dimensions using a linear combination of the 3 basis vectors.  

    By default,   i , j and k are the basis vectors for us.

  Now for a given position vector or coordinate vector, P [p, q, r], we want to change the basis.  The new basis is vectors:
             V1 =[ a,b,c]     V2 = [d, e, f].    V3 = [g , h , k]

     Find constants α, β and λ such that :
             [p, q, r ] = α [a, b, c ] + β [ d, e, f ] + λ [ g, h, k ]

         So p = αa + βd + λ g
             q = αb + β e + λ h
              r = α c + β f + λ k

   Solve the three above equations to find α , β and λ.

The answer will be then:  [ α, β, λ ].    ANSWER