Let $\displaystyle a,b,x\in\mathbb{R}$ not all equal to zero and $\displaystyle X=\{(x,y,z)\in\mathbb{R}^3: ax+by+cz=0\}$

Find a subspace Y such that $\displaystyle \mathbb{R}^3= X\oplus Y$ and deduce X is 2 dimensional.

I am lost on this one.

Note:

$\displaystyle \oplus$ is the external direct sum define as $\displaystyle X\oplus Y=(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$

$\displaystyle \lambda (x_1,y_1)=(\lambda x_1,\lambda x_2)$

If $\displaystyle X \cap Y =\{0\}$, then $\displaystyle \oplus$ can be define as $\displaystyle \text{dim}(X+Y)=\text{dim}X+\text{dim}Y$ that is the internal direct sum $\displaystyle X\oplus Y$