Find a unit vector perpendicular to `A=2hat(i)+3hat(j)+hat(k)` and
`B=hat(i)-hat(j)+hat(k)` both.

Options

A. `(1)/(sqrt2)(21hat(i)-3hat(j))`
B. `(1)/(sqrt(5))(4hat(i)+hat(j)+5hat(k))`
C. `(1)/(sqrt(42))(4hat(i)-hat(j)-5hat(k))`
D. `(1)/(sqrt(42))(4hat(i)-hat(j)+5hat(k))`

Explanation

`hat(n)=(C)/(C)=(AxxB)/(|AxxB|)`
`Here, AxxB=|{:(hat(i),hat(j),hat(k)),(2,3,1),(1,1,1):}|=hat(i)(3+1)+hat(j)(1-2)+hat(k)(-2-3)`
`=4hat(i)-hat(j)-5hat(k)`
Further, `|AxxB|=sqrt((4)^(2)+(-1)^(2)+(-5)^(2))=sqrt(42)`
:. The desired unit kvector is `hat(n)=(AxxB)/(|AxxB|)`
or `hat(n)=(1)/(sqrt(42))(4hat(i)-hat(j)-5hat(k))`