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

1 Answer
Feb 15, 2017

The unit vector is =<-10/sqrt117,1/sqrt117,4/sqrt117>

Explanation:

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

Let veca=<1,-2,3>

vecb=<1,2,2>

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

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

=hati(-10)-hatj(-1)+hatk(4)

=<-10,1,4>

Verification

veca.vecc=<1,-2,3>.<-10,1,4>=-10-2+12=0

vecb.vecc=<1,2,2>.<-10,1,4>=-10+2+8=0

The modulus of vecc=||vecc|=||<-10,1,4>||=sqrt(100+1+16)=sqrt117

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

=1/sqrt117<-10,1,4>