How can I find a constant c such that norm(c*v1 - v2) is as close to 0 as possible?

Question

I have two vectors, they are nearly linearly dependent. I would like to normalize them in a least squared sense. How can I do this?

Answer 1

The square of the norm of c*v_1 -v_2 is the inner product

=c^2 -2c+.
This is a quadratic in c so is minimized when
c=/

Answer 2

Project vector v2 on v1. The projection is the c*v1 you are looking for.

Algebraic derivation: minimize the squared norm.

[1] ... |c*v1 - v2|^2 is minimal
[2] ... c^2|v1|^2 - 2c(v1.v2) + |v2|^2 is minimal

The minimum of ax^2 + bx + c is found when x = -b/2a; here:

[3] ... c = (v1.v2) / |v1|^2

Answer 3

c*v1 -v2 Set this to 0
c*v1 -v2 =0 Add v2 to both sides
c*v1 = v2 Now divide both sides by v1
c = v2/v1 Now plug this value into c and when you do the original problem it should equal 0.

Answer 4

write it out (this depends on your norm i use euclidean)
: r(c) = (cx1-x2)^2 +(cy1 - y2)^2

v1 =(x1,y1), v2=(x2,y2) all given.
r(c) is a parabole minimize for c.