If a=i+3j+k and b=i+12j+k, find a unit vector with positive first coordinate orthogonal to both a and b.

The cross product of the vectors will be orthogonal to both:
axb
[tex]=<1,3,1>x<1,12,1>
=
i j k
1 3 1
1 12 1
=<3-12, -1+1, 12-3>
=<-9,0,9>
Since it is required to have positive first coordinate, we reverse the sign of the components, namely reverse the direction of the vector, which is still orthogonal to both a and b:
Required vector is <9,0,-9> which is orthogonal to both a and b.