Let X be a real inner product space and let x notequalto 0 a

Let X be a real inner product space and let x notequalto 0 and y notequalto 0 be two vectors in X. Find min_alpha epsilon R ||x + alpha y||.

Solution

Since x and y are two non zero vectors in a real inner product space X we got to find the minimum value of || x+ ay||
For minimum value of modulous we know that it is equal to zero; But that happens when x and y are zero thus in this case teh value will be limiting equal to zero from the positive side as x and y can be chosen to be the smallest of the vectors.
In another case, x can be chsen to be a vector in the opposite dierction of y but equal magnitude, in that case upon a being = 1; the modulous becomes || x + ay || = || x + (1) (-x) || = || x - x || = 0