If `vec(a) = (hat(i) - 2 hat(j)+ 3 hat(k)) and vec(b) = (2 hat(i) + 3 hat(j)- 5 hat(k))` then find `(vec(a) xx vec(b)) and " verify that " (vec(a) xx vec(b))` is perpendicular to each one of` vec(a) and vec(b)`.

Explanation

We have `(vec(a) xx vec(b))= |(hat(i), hat(j), hat(k)),(1,-2,3),(2,3,-5)|`
`=(10-9) hat(i)-(-5-6)hat(j) + (3+4) hat(k) = ( hat(i) + 11 hat(j) + 7 hat(k))`.
Now, `(vec(a) xx vec(b) ) * vec(a)= (hat(i) + 11 hat(j) + 7 hat(j))*(hat(i) - 2hat(j) + 3 hat(k)).`
`:. (vec(a) xx vec(b))*vec(a) = (1-22+21)=0`
`:. (vec(a) xxvec(b)) bot vec(a).`
And,`(vec(a) xx vec(b))* vec(b)=(hat(i) +11vec(j)+7vec(k))*(2hat(i)+3 hat(j)-5hat(k))`
`=(2+33-35)=0`.
`:. (vec(a)xxvec(b))bot vec(b)`.