Find the coordinates of a unit vector that is perpendicular to both of the vectors

x=(1,1,1) and y=(1,3,-1) and makes an obtuse angle with the z-axis.

x = i + j + k

y = i + 3j - k

vector perpendicular to both is given by (yXx) ..let z = (yXx) then

Z = (i+3j-k)x(i+j+k)

Z = 4i - 2j  - 2k

unit vector = Z/modZ = (4i-2j-2k)/root24

x,y,z components are (4/root24 , -2/root24 , -2/root24 )

let z be the required vector. z=±(x×y)/l(x×y)l, So there are two vectors. Consider their dot products with k, one of them will be positive and the other will be negative. The one that is negative makes an obtuse angle with the z-axis. This vector is the answer.