Question #210434

Find an expression for 12

jj~u + ~vjj2 + 12

jj~u 􀀀 ~vjj2 in terms of jj~ujj2 + jj~vjj2.

(3.2) Find an expression for jj~u + ~vjj2 􀀀 jj~u 􀀀 ~vjj2 in terms of ~u  ~v (3)

(3.3) Use the result of (3.2) to deduce an expression for jj~u + ~vjj2 whenever ~u and ~v are orthogonal (1)

to each other.


1
Expert's answer
2021-06-25T11:45:03-0400

(3.1)


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

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

=u2+2(u,v)+v2=||\vec u||^2+2(\vec u, \vec v)+||\vec v||^2


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

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

=u22(u,v)+v2=||\vec u||^2-2(\vec u, \vec v)+||\vec v||^2




12u+v2+12uv2\dfrac{1}{2}||\vec u+\vec v||^2+\dfrac{1}{2}||\vec u-\vec v||^2

=12(u2+2(u,v)+v2)+12(u22(u,v)+v2)=\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)

=u2+v2=||\vec u||^2+||\vec v||^2

(3.2)


u+v2uv2||\vec u+\vec v||^2-||\vec u-\vec v||^2

=u2+2(u,v)+v2(u22(u,v)+v2)=||\vec u||^2+2(\vec u, \vec v)+||\vec v||^2-(||\vec u||^2-2(\vec u, \vec v)+||\vec v||^2)

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

(3.3)

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


u+v2=u2+v2=uv2||\vec u+\vec v||^2=||\vec u||^2+||\vec v||^2=||\vec u-\vec v||^2


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