Question

Write the value of `vec a .(vecb xxveca)`

Answer 1

Let:

`vec a=a_1hati+a_2hatj+a_3hatk`

`vecb=b_1hati+b_2hatj+b_3hatk`

`therefore veca.(vecb xx veca)`

`=(a_1hati+a_2hatj+a_3hatk).[(b_1hati+b_2hatj+b_3hatk) xx (a_1hati+a_2hatj+a_3hatk)]`

`=(a_1hati+a_2hatj+a_3hatk).[(b_2a_3-b_3a_2)hati-(b_1a_3-b_3a_1)hatj+(b_1a_2-b_2a_1)hatk]`

`=(b_2a_3-b_3a_2)a_1-(b_1a_3-b_3a_1)a_2+(b_1a_2-b_2a_1)a_3`

`=a_1a_3b_2−a_1a_2b_3−a_2a_3b_1+a_1a_2b_3+a_2a_3b_1−a_1a_3b_2`

Alternate Method:

`vec b xx vec a` is a vector perpendicular to both `veca and vecb`

`∴ vecb xx veca ⊥ veca` and `vecb xx veca ⊥ vecb`

⇒ `veca . (vecb xx veca)=0`