$($a$)$ Given three vectors $a=+4j,b=-2i+yj$ and $c=2i+2j.$ If $a+c$ perpendicular to $b-c$ and $a*b=12,$ determine the values of $x$ and $y.$

Ans : $x=-2,y=2$

$($b$)$ Three points $A,B$ and $C$ have positon vectors $2i+j+k,3i-j+3k$ and $i+3j+pk$ respectively. Find the value of $p$ such that $A,B$ and $C$ are collinear. Hence, find the ratio of $CB$:$AB.$

Ans: $p=-1,CB$:$AB=2$ :$1$