An example of sum that is not direct

Question

Give an example of $V_1, V_2, V_3\subset W$ where $V_i\cap V_j=\{0\}$ for every $i\ne j$ yet the sum $V_1+V_2+V_3$ is not direct.

My answer: Let $W=\mathbb{R}^3$, $V_1=\{(x,2x,0):x,y\in\mathbb{R}\}$, $V_2=\{(x,3x,0):x,y\in\mathbb{R}\},V_3=\{(x,4x,0):x,y\in\mathbb{R}\}$. All are subspaces, but $V_1+V_2+V_3\ne\mathbb{R}^3$ hence it is not a direct sum.

Am I correct?

Answer 1

I think you should take three lines passing through $(0,0)$ in $\mathbb{R}^2$.