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`.

Explanation

(a) Component of `vecA` along `vecB=((vecA.vecB)/(B))hatB=((vecA.vecB)/(B))(vecB)/(B)=[((3hati+hatj).(hatj+2hatk))/sqrt(5)]((hatj_2hatk))/sqrt(5)=(1)/(5)(hatj+2hatk)`
(b) Area of the parallelogram `=|vecAxxvecB|=|{:(hati,hatj,hatk),(3,1,0),(0,1,2):}|=|2hati-6hatj+3hatk|=sqrt(2^(2)+(-6)^(2)+3^(2))=7 "units"`
(c) Unit vector perpendicular to both `vecA & vecB hatn=(vecAxxvecB)/(|vecAxxvecB|)=(2hati-6hatj+3hatk)/(7)=(2)/(7)hati-(6)/(7)hatj+(3)/(7)hatk`