There are two vectors `vecA=3hati+hatj` and `vecB=hatj+2hatk`. For these two vectors-
(a) Find the component of `vecA` along `vecB` in vector form.
(b) If `vecA & vecB` are the adjacent sides of a parallalogram then find the magnitude of its area.
(c) Find a unit vector which is perpendicular to both `vecA & vecB`.

Answer

(i) `1/5 (hat(j)+2hat(k)), 3hat(i)+4/5hat(j)-2/5hat(k)` (ii) 7 units (iii) `2/7hat(i)-6/7hat(j)+3/7 hat(k)`

Explanation

(i) component of `vec(A)` along `vec(B)=((vec(A).vec(B))/B)hat(B)`
`=((vec(A).vec(B))/B)vec(B)/B=[((3hat(i)+hat(j)).(hat(j)+2hat(k)))/sqrt(5)]((hat(j)+2hat(k)))/sqrt(5)=1/5 (hat(j)+2hat(k))`
Component of `vec(A) bot vec(B)`
`=vec(A)-[(vec(A).vec(B))/vec(B)]hat(B)=3hat(i)+hat(j)-[1/5(hat(j)+2hat(k))]`
(ii) Area of the parallelogram
`=|vec(A)xxvec(B)|=|(hati,hatj,hatk) ,(3,1,0),(0,1,2)|=|2hati-6hatj+3hatk|`
`=sqrt(2^(2)+(-6)^(2)+3^(2))=7` units
(iii) Unit vector perpendicular to both `vec(A)` & `vec(B)`
`hat(n)=(vec(A)xxvec(B))/(|vec(A)xxvec(B)|)=(2hati-6hatj+3hatk)/7=2/7hat(i)-6/7 hat(j)+3/7 hat(k)`