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 `(OA) and (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)xxvec(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)`
`=1ohat(i)+10hat(j) +15hat(k)`
`rArr |vec(a)xxvec(b)|=sqrt(10^(2)+10^(2)+15^(2))=sqrt(425)=5sqrt(17)`
(i) Area of `DeltaOAB=(1)/(2)|vec(a)xxvec(b)|=(1)/(2).5sqrt(17)sq`.units
`(5)/(2)sqrt(17)sq`. units
(ii)Area of parallelogram fromed by OA and OB as
adjacent sides `=|vec(a)xxvec(b)|=5sqrt(17)sq`. units.