Question

Find a unit vector that is orthogonal to both u and v. u = −8, −6, 4 v = 10, −18, −2

Answer 1

A vector w orthogonal to u and v can be obtained by the cross product of u and v, namely u x v.  Subsequently, u x  v may be reduced to a unit vector.

u<-8,-6,4>
v<10,-18,-2>

w=u x v 
=
  i      j   k
 -8   -6  4
10 -18 -2
=<12+72, 40-16, 144+60>
=<84, 24, 204>
=12 < 7,2,17>

Check that w is orthogonal to u and v
w.u = 12<7,2,17><-8,-6,4> = 12(-56-12,68)=0
w.v = 12<7,2,17><10,-18,-2> = 12(70-36-34)=0
Therefore w is orthogonal to both u and v.

Unit vector of w
$\frac{}{\sqrt{7^2+2^2+17^2}}=\frac{1}{3\sqrt{38}}$