Question

Find a unit vector that is orthognal to both [1,1,0] and [1,0,1]

Answer 1

`(-(- √(3))/(3),-(√(3))/(3) , (√(3))/(3))`

Explanation:

take u = [1,1,0] and

v = [1,0,1]

calculate the cross product of u and v

u × v = [1,1,0] × [1,0,1]

`[\begin{vmatrix}0  & 1\\  1 &1\end{vmatrix}\begin{vmatrix}1  & 1\\ 1 & 0\end{vmatrix}\begin{vmatrix}1  & 0\\ 0 & 1 \end{vmatrix} ]`

= (-1,-1,1)

then

`\left \| -1,-1,1 \right \| = √( -1^2 + -1^2 +1)= √(3)`

`(1)/(√(3))(-1,-1,1 )` to unit vector by dividing by
`√(3)`

`((1)/(√(3))(-1,-1,1 ))/(√(3)) = (-(- √(3))/(3),-(√(3))/(3) , (√(3))/(3))`