# Simple exercise regarding space spanned by two vectors

Consider the vectors $X_1=(1,3,2)$ and $X_2=(-2,4,3)$ in $\mathbb R^3$. Show that the set spanned by $X_1$ and $X_2$ is given by $\{(Y_1,Y_2,Y_3):Y_1-7Y_2+10Y_3=0\}$.

• Can you please show what you have tried so far? This seems like a standard homework exercise. – air Sep 3 '15 at 13:44
• I think the question is wrong because- $\alpha-2\beta=1$, $3\alpha+4\beta=-7$ and $2\alpha+3\beta=10$. These system of equations does not have any solution. – Rajat Sep 3 '15 at 13:58

$$\vec n=X_1\times X_2=(1,-7,10).$$
All vectors in the spanned plane will be perpendicular to $\vec n$. So, the equation is $$\vec v\cdot \vec n =0.$$
Or, if $\vec v=(x,y,z)$ then the equation is
$$x-7y+10z=0.$$