Given three vectors $\vec a=6\hat i-3\hat j,\,\vec b=2\hat i-6\hat j$ and $\vec c=-2\hat i+21\hat j$ such that $\vec α=\vec a+\vec b+\vec c$. Then the resolution of the vector $\vec α$ into components with respect to $\vec a$ and $\vec b$ is given by:

$2\vec a-3\vec b$

$\vec α=\vec a+\vec b+\vec c=6\hat i +12\hat j$

so component along $\vec a$ and $\vec b$

so $\vec α=p\vec a+q\vec b$

so $(6p+2q)\hat i+(-3p-6q)\hat j=6\hat i +12\hat j$

$p=2,q=-3$

$\vec α=2\vec a-3\vec b$