[Math] Given the following vectors $\vec u$ and $\vec v$, find a vector $\vec w$ in $\mathbb R^3$ so that ${\vec u, \vec v, \vec w}$ spans $\mathbb R^3$.

Question

Given the following vectors $\vec u$ and $\vec v$, find a vector $\vec w$ in $\mathbb R^3$ so that $\{\vec u, \vec v, \vec w\}$ spans $\mathbb R^3$ and a non-zero vector $\vec z$ in $\mathbb R^3$ so that $\{\vec u, \vec v, \vec z\}$ does not span $\mathbb R^3$.

I know that $\vec u=[-9\ -10\ -1]$ and $\vec v=[8\ -4\ \ 1]$.
I was thinking of making a matrix and trying to solve it to get the identity matrix.
I am not really sure how to approach this.

Answer 1

For $w$ you can take $w=[1,0,0]$. Then the determinant of the matrix $A$ with rows $u,v$ and $w$ is $-14$, so $A$ has full rank.

For $z$ you may take $z=u$, so that $u,v,z$ are linearly dependent. Or $z=u+v$