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

1 Answer
Feb 20, 2017

Answer:

The unit vector is #=<5/sqrt42,4/sqrt42,1/sqrt42>#

Explanation:

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

Let #veca=<-1,1,1>#

#vecb=<1,-2,3>#

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

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

#=hati(5)-hatj(-4)+hatk(1)#

#=<5,4,1>#

Verification

#veca.vecc=<-1,1,1>.<5,4,1>=-5+4+1=0#

#vecb.vecc=<1,-2,3>.<5,4,1>=5-8+3=0#

The modulus of #vecc=||vecc||=||<5,4,1>||=sqrt(25+16+1)=sqrt42#

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

#=1/sqrt42<5,4,1>#