Hi Riemann.
Let \(\displaystyle{(x,y,z)\in W_1\cap W_2\cap W_3}\). Then,
\(\displaystyle{(x,y,z)\in W_1\implies x+yz=0\,\,(I)}\)
\(\displaystyle{(x,y,z)\in W_2\implies 3\,x+y2\,z=0\,\,(II)}\)
\(\displaystyle{(x,y,z)\in W_3\implies x7\,y+3\,z=0\,\,(III)}\).
The relations \(\displaystyle{(I)\,,(II)}\) give us
\(\displaystyle{x+yz=3\,x+y2\,z\iff z=2\,x}\) and then the relation \(\displaystyle{(III)}\) becomes
\(\displaystyle{x7\,y+6\,x=0\iff 7\,x7\,y=0\iff x=y}\)
So, \(\displaystyle{(x,y,z)=(x,x,2\,x)=x\,(1,1,2)\in\langle{(1,1,2)\rangle}}\).
Conversely, if \(\displaystyle{(x,x,2\,x)\in\langle{(1,1,2)\rangle}}\), then,
\(\displaystyle{x+x2\,x=0\implies (x,x,2\,x)\in W_1}\)
\(\displaystyle{3\,x+x2\,2\,x=4\,x4\,x=0\implies (x,x,2\,x)\in W_2}\)
\(\displaystyle{x7\,x+3\,2\,x=7\,x7\,x=0\implies (x,x,2\,x)\in W_3}\)
which means that \(\displaystyle{(x,x,2\,x)\in W_1\cap W_2\cap W_3}\)
So,
\(\displaystyle{W_1\cap W_2\cap W_3=\langle{(1,1,2)\rangle}\implies \dim_{\mathbb{R}}(W_1\cap W_2\cap W_3)=1}\)
Similarly, \(\displaystyle{W_1\cap W_2=\langle{(1,0,2)\,,(0,1,0)\rangle}}\) and
\(\displaystyle{\dim_{\mathbb{R}}(W_1\cap W_2)=2}\).
