Enforce the norm of the addition of two vectors plus scalar

Hi,

I have two vectors of the same dimension: and and a scalar β. I'm looking for the value of β such that .

Do you have an idea how to find β ?

Thanks.

Answer 1

[Mohammad Alhashash]

I assumed that a and b are 3 elements-vector and then solved the equation beta analytically.

a = [1 2 3];
b = [2 3 4];
beta = roots([b(1)^2+b(2)^2+b(3)^2 , 2*(a(1)*b(1)+a(2)*b(2)+a(3)*b(3)),a(1)^2+a(2)^2+a(3)^2-1])

the result is two solution corresponding to two different value of beta.

to verify our solution, simply try it;

norm(a+beta(1)*b)

or :

norm(a+beta(2)*b)

[Thomas Barbier]

Thanks.

This solution exists though if , otherwise no (real) solution is available.