Find the coordinates of the vector x relative to the basis b
Answers
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
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