question about inner product formula

[ran_out]

hi all,

i have a question regarding the relation:

(x_transpose)y= norm_x* norm_y * cos(α)

where x, y are vectors and α is the angle between them.

from law of cosines u get:

cos(α) = (norm_ x^2 + norm y ^2 - norm_(x-y)^2) / (2*norm_x*norm_y)

right?

but how do you get the relation:

norm_ x^2 + norm y ^2 - norm_(x-y)^2 = 2*(x_transpose)y

which is needed to get the final result

(x_transpose)y = norm_x* norm_y * cos(α)

Thanx for help

 

[steve10]

According to the definition of the inner product,
norm_(x-y)^2=(x-y,x-y)
=norm_x^2-xy-yx+norm_y^2
=norm_x^2 + norm_y^2 - 2*(x_transpose)y

Therefore,
norm_ x^2 + norm y ^2 - norm_(x-y)^2
=2*(x_transpose)y.

 

[ran_out]

huge thanx for that