If vectors 2 i-j+k, i+2 j-3 k and 3 i+aj+5 k are coplanar, then the...

If given vectors are coplanar, then there exists two scalar quantities $\mathrm{x}$ and $\mathrm{y}$ such that $2 \hat{i}-\hat{j}+\hat{k}=x(\hat{i}+2 \hat{j}-3 \hat{k})+y(3 \hat{i}+a \hat{j}+5 \hat{k})$ Comparing coefficient of $\hat{i}, \hat{j}$ and $\hat{k}$ on both sides of (1) we get $x+3 y=2,2 x+a y=-1,-3 x+5 y=1 \ldots(2)$ Solving first and third equations, we get $\mathrm{x}=1 / 2, \mathrm{y}=1 / 2$ Since the vectors are coplanar, therefore these values of $x$ and $y$ will satisfy the equation $2 x+a y=-1\therefore 2(1 / 2)+a(1 / 2)=-1 \Rightarrow a=-4$