Answer
See below
Work Step by Step
We are given: $u=(2,3)\\
v=(4,-1)$ in $R^2$
Let $\theta$ be the angle between $u$ and $v$.
We obtain:
$\cos \theta=\frac{}{||u||.||v||}\\
=\frac{2.2.4+3.(-1)}{\sqrt 2.2.2+3.3.\sqrt 2.4.4+(-1).(-1)}\\
=\frac{13}{\sqrt 17.\sqrt 33}\\
=\frac{13\sqrt 561}{561}$
$\rightarrow \theta=\arccos(\frac{13\sqrt 561}{561})$
Consider $u=(-2,-1,2,4)\\
v=(-3,5,1,1)$ in $R^4$
We have:
$\cos \theta=\frac{}{||u||.||v||}\\
=\frac{2.(-2).(-3)+(-1).5+2.1+4.1}{\sqrt 2.(-2).(-2)+(-1)(-1)+2.2+4.4.\sqrt 2.(-3).(-3)+5.5+1.1+1.1}\\
=\frac{13}{\sqrt 29.\sqrt 45}\\
=\frac{13\sqrt 145}{435}$
$\rightarrow \theta=\arccos(\frac{13\sqrt 561}{561})$
