Answer to Question #209776 in Analytic Geometry for Jaguar

Question

(3.1) Find an expression for 1

2

||~u + ~v||2 +

1

2

||~u − ~v||2

in terms of ||~u||2 + ||~v||2

.

(3.2) Find an expression for ||~u + ~v|| 2 − ||~u − ~v||2

in terms of ~u · ~v

(3.3) Use the result of (3.2) to deduce an expression for ||~u + ~v||2 whenever ~u and ~v are orthogonal

to each other.

Answer 1

(3.1)

$(\vec u+\vec v)^2=(\vec u+\vec v)\cdot(\vec u+\vec v)$

$=(\vec u, \vec u)+2(\vec u, \vec v)+(\vec v, \vec v)$

$=||\vec u||^2+2(\vec u, \vec v)+||\vec v||^2$

$(\vec u-\vec v)^2=(\vec u-\vec v)\cdot(\vec u-\vec v)$

$=(\vec u, \vec u)-2(\vec u, \vec v)+(\vec v, \vec v)$

$=||\vec u||^2-2(\vec u, \vec v)+||\vec v||^2$

$\dfrac{1}{2}||\vec u+\vec v||^2+\dfrac{1}{2}||\vec u-\vec v||^2$

$=\dfrac{1}{2}(||\vec u||^2+2(\vec u, \vec v)+||\vec v||^2)+\dfrac{1}{2}(||\vec u||^2-2(\vec u, \vec v)+||\vec v||^2)$

$=||\vec u||^2+||\vec v||^2$

(3.2)

$||\vec u+\vec v||^2-||\vec u-\vec v||^2$

$=||\vec u||^2+2(\vec u, \vec v)+||\vec v||^2-(||\vec u||^2-2(\vec u, \vec v)+||\vec v||^2)$

$=4(\vec u, \vec v)$

(3.3)

If $\vec u \perp \vec v,$ then $(\vec u, \vec v)=0$

$||\vec u+\vec v||^2=||\vec u||^2+||\vec v||^2=||\vec u-\vec v||^2$