TITLE: Have the answer for a vectors question but do not understand how to arrive at it
QUESTION [0 upvotes]: Question : Find the value of x so that A[-1,3,4] , B[-2,3,-1] and C[-5,6, x] all lie on a plane that contains the origin 
Answer : -7
The confusion for me here is more so what the question is looking for /  the method it wants me to take in order to arrive at the desired answer, I would simply turn to a solutions manual or a service like Photomath however the former does not show process and the latter only works with numerical problems

REPLY [0 votes]: the method it wants me to take in order to arrive at the desired answer

This might depend.  Sometimes, homework assignment force a specific way or approach to solve a problem. Sometimes to get familiar with a specific topic, sometimes not to bust a teacher's head.
In your case, the statement is that $A$, $B$ and $C$ are not linear independent, i.e. their equation 
$$\alpha A + \beta B +\gamma C = 0$$
has solution(s) in real numbers $\alpha$, $\beta$ and $\gamma$ where not all of them are zero.  One way to find this is to form a $3\times 3$ matrix of the vectors (does not matter whether you use them for columns or rows) and solve
$$\det M=0$$
You can also understand this as linear system of 3 equations with 3 unknowns $\alpha$, $\beta$ and $\gamma$ that has to be solved dependent on a parameter $x$, i.e. $\alpha$, $\beta$ and $\gamma$ are actually functions of $x$, and you are searching $x$'s that permit non-trivial solution(s).
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