<s>[INST] find range for dependent 
There are three vector $a$, $b$, $c$ in three-dimensional real vector space, and the inner product between them $a\cdot a=b\cdot b = a\cdot c= 1, a\cdot b= 0, c\cdot c= 4$. When setting $x = b\cdot c$, answer the following question: when $a, b, c$ are linearly dependent, find all possible values ​of $x$.
  (dot here means dot product)

For dependent condition 
$$\begin{align}
(a×b)·c &= 0\\
a·(b×c) &= 0\\
a(bc \sin θ)&=0
\end{align}$$
So $\theta = 0, \pi$.
Then
$$\begin{align}
x&=|b||c| \cos \theta \\
x&=2 \cos \theta \\
\implies x &= 2 \cos 0, x = 2 \cos \pi \\
x &= \mp 2
\end{align}$$
Am I right using triple cross product? or should i find $\theta$?

 [/INST] For the first identity, $a \times b$ is orthogonal to both $a,b$, hence $c$ must be in the plane spanned by the orthonormal $a,b$.
So we can write $c$ as 
$$
c = \alpha a + \beta b
$$
plugging this in we see that
$$
1 = a \cdot c = \alpha a \cdot a = \alpha
$$
and 
$$
x = b \cdot c = \beta b \cdot b = \beta 
$$
hence the norm squared of $c$ is
$$
c\cdot c = \alpha^2 + \beta^2 = 1 + x^2 = 4
$$
this gives
$$
x = \pm \sqrt{3}
$$
</s>