Solution: (a) Given that the vector -2 hat(i)+9 hat(j)" Now "{:[|-2 hat(i)+9 hat(j)|=sqrt((-2)^(2)+(9)^(2))],[=sqrt(4+81)=sqrt85.]:}since vec(a)=x_(i)+y hat(j) then | vec(a)|=sqrt(x^(2)+y^(2)).:. The unit vectors in the same livection of -2 hat(i)+9 hat(j) is quad(-2( hat(i))+9( hat(j)))/(1-2( hat(i))+9( hat(j)))=(-2( hat(i))+9( hat(j)))/(sqrt85)Hence vec(u)=(-2( hat(i))+9( hat(j)))/(sqrt85)(b) Given that the vector 12 hat(i)-8 hat(j)+8 hat(k).{:[:.|12 hat(i)-8 hat(j)+8 hat(k)|=sqrt((12)^(2)+(-8)^(2)+8^(2))],[=sqrt(144+64+64)=sqrt272=4sqrt17.]:}Hence the required unit vector oppositely directed to 12 hat(i)-8 hat(j)+8 hat(k) is-(12( hat(i) ... See the full answer