Determine $\operatorname{Ker}$ of a linear application

Question

Let $\underline v$ and $\underline w$ two non parallel vectors and $T:E_3\longrightarrow E_3$ defined by $$ T(\underline x)=(\underline x\cdot\underline v)\underline v+(\underline x\cdot\underline w)\underline w. $$ Then

1. $T$ is linear and $\operatorname{Ker}T$ is formed by all vectors parallel to $\underline v\times \underline w$;
2. No one of other answers;
3. $T$ is not linear but injective;
4. $T$ is not linear but surjective;
5. $T$ is linear and $\operatorname{Ker}T$ is formed by all vectors parallel to $\underline v+\underline w$.

My attempt. $T$ is clearly linear. So $3$ and $4$ are false. But since $T(\underline x)=\underline 0$ if and only if $\underline x=\underline 0$ or $\underline x$ is orthogonal to $\underline v$ and $\underline w$, I conclude that answer $2$ is the correct one.

Is my attempt right?

Answer 1

No. The correct answer is 1. Suppose that $u\in\ker T$, that is, that $T(u)=0$. Then $u.v=u.w=0$. In other words, $u$ is orthogonal to both $u$ and $v$. And this is the same thing as asserting that $u$ is parallel to $v\times w$.