The vectors from origin to the points A and B are `vec(a)=3hat(i)-6hat(j)+2hat(k) and vec(b)= 2hat(i) +hat(j)-2hat(k)` respectively. Find the area of :
(i) the triangle OAB
(ii) the parallelogram formed by `vec(OA) and vec(OB)` as adjacent sides.

Explanation

Given `vec(OA)=vec(a) =3hat(i)-6hat(j)+2hat(k) and vec(OB)=vec(b)=2hat(i)+hat(j)-2hat(k)`
`:. (vec(a))xx(vec(b))=|(hat(i), hat(j),hat(k)),(3, -6, 2), (2, 1, -2)|`
`=(12-2) hat(i)-(-6-4)hat(j)+(3+12)hat(k)`
`=10hat(i)+10hat(j)+15hat(k)`
`rArr|vec(a)xxvec(b)|=sqrt(10^(2)+10^(2)+15^(2))=sqrt(425)=5sqrt(17)`
Area of `OAB=(1)/(2)|vec(a)xxvec(b)|=(1)/(2).5sqrt(17)sq.units`
`=(5)/(2)sqrt(17)sq. units`
(ii) Area of parallelogram formed by `vec(OA) and vec(OB)` as adjacent sides `=|vec(a)xxvec(b)|=5sqrt(17)`sq. units.