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Let $v_1, v_2 \in \mathbb{R}^3$, such that {$v_1, v_2$} is linearly independent. If $w$ is orthogonal to $v_1$ and $v_2$, then is {$v_1, v_2, w$} linearly independent?

There are two cases, $w = \overrightarrow{0}$ is $w$ is not a zero vector.

Case: $w$ is not a zero vector, then the set {$v_1, v_2, w$} is linearly independent.

**Case: ** But is $w$ is a zero vector, then {$v_1, v_2, w$} is NOT linearly independent.

So what is the verdict then?

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Strictly speaking, what you have said gives the correct answer: sometimes $\{v_1,v_2,w\}$ is independent, and sometimes it isn't.

However probably (check with your teacher) the intended question was

Let $v_1, v_2 \in \mathbb{R}^3$, such that {$v_1, v_2$} is linearly independent. Is it always true that if $w$ is orthogonal to $v_1$ and $v_2$, then {$v_1, v_2, w$} is linearly independent?

- in which case the answer is "no", not always, because of your second example.

Or perhaps

Let $v_1, v_2 \in \mathbb{R}^3$, such that {$v_1, v_2$} is linearly independent. If $w\ne0$ and $w$ is orthogonal to $v_1$ and $v_2$, then is {$v_1, v_2, w$} linearly independent?

- in which case the answer is "yes".

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    +1, But I think in some textbooks the word "orthogonal" is only used with non-zero vectors. I'm not saying that this is good or correct, but the fact of the matter is that some authors do make this assumption even when they forget to spell it out. So this is indeed a matter of interpreting the question, and the OP's best bet is to ask the teacher. – 2017-01-24