Someone please explain cosine similarity equation to me? I'm trying to understand the cosine similarity in a simple and graphical way, very much similar to this question here but I do not understand how the person got to their answer.

The vectors are $(x_1 \  x_2 \  x_3 \  x_4 \  x_5 \   x_6 \  x_7 \  x_8)^T=(2 \  0 \  1 \  1 \  0 \  \  2 \  1 \  1)^T$ and $(y_1 \  y_2 \  y_3 \  y_4 \  y_5 \   y_6 \  y_7 \  y_8)^T=( \ 2 \  1 \  1 \  0 \  1 \  1 \  1 \  1)^T$  
The cosine of angle ($\alpha$) between the two vectors  is $cos(\alpha)=\large{\frac{\sum_{i=1}^8 x_i\cdot y_i}{\sqrt{\sum_{i=1}^8 x_i^2}\cdot \sqrt{\sum_{i=1}^8 y_i^2}}}=\frac{2\cdot 2+0 \cdot 1+ 1\cdot 1+ 1\cdot 0 + 0 \cdot 1+ 2\cdot 1+ 1\cdot 1+ 1\cdot 1}{\sqrt{2^2+0^2+1^2+1^2+0^2+2^2+1^2+1^2}\cdot \sqrt{2^2+1^2+1^2+0^2+1^2+1^2+1^2+1^2}}$
$=\frac{4+1+2+1+1}{\sqrt{4+1+1+4+1+1}\cdot \sqrt{4+1+1+1+1+1+1}}=\frac{9}{\sqrt{12}\cdot \sqrt{10}}=0.8216$
Thus the equation is $\cos(\alpha)=0.8216$
$\alpha=arcos(0.8216)=34.76°$
