If $(x,y,z)\ne (\mathrm{0,0},0)$ and

$(\bar{i}+\bar{j}+3\bar{k})x+(3\bar{i}-3\bar{j}+\bar{k})y+(-4\bar{i}+5\bar{j})z=a(x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k})$ 

 

Then the value of a is_____

The given vector equation can be written as

$\begin{array}{rr} & \left[\right(1-a)x+3y-4z]\overrightarrow{i}[x-(3+a)y+5z]\overrightarrow{j}+[3x+y-az]\overrightarrow{k}=0\\ & \Rightarrow (1-a)x+3y-4z=0\\ & x-(3-a)+5z=0\text{ and }3x+y-az=0\end{array}$The above system of equation has a non trivial solution.

$\begin{array}{rr} & \Rightarrow \left|\begin{array}{ccc}1-a& 3& -4\\ 1& -(3-a)& 5\\ 3& 1& -a\end{array}\right|=0\\ & \Rightarrow a=0,-1\end{array}$