Question

What is the unit vector that is normal to the plane containing `(- 3 i + j -k)` and #(- 2i - j - k)?

Answer 1

The unit vector is `=<-2/sqrt30,-1/sqrt30,5/sqrt30>`

Explanation:

We calculate the vector that is perpendicular to the other 2 vectors by doing a cross product,

Let `veca=<-3,1,-1>`

`vecb=<-2,-1,-1>`

`vecc=|(hati,hatj,hatk),(-3,1,-1),(-2,-1,-1)|`

`=hati|(1,-1),(-1,-1)|-hatj|(-3,-1),(-2,-1)|+hatk|(-3,1),(-2,-1)|`

`=hati(-2)-hatj(1)+hatk(5)`

`=<-2,-1,5>`

Verification

`veca.vecc=<-3,1,-1>.<-2,-1,5>=6-1-5=0`

`vecb.vecc=<-2,-1,-1>.<-2,-1,5>=4+1-5=0`

The modulus of `vecc=||vecc||=||<-2,-1,5>||=sqrt(4+1+25)=sqrt30`

The unit vector `= vecc /(||vecc||)`

`=1/sqrt30<-2,-1,5>`