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 $(\vec u, \vec v)=0$

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