### Instruction: Find a unit vector that is orthogonal to both $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}.$.### Response: A unit vector is a vector with length 1, and two vectors are orthogonal if their dot product is zero.
To find a vector that is orthogonal to both given vectors, I can use the cross product, which gives a vector that is perpendicular to the plane spanned by the two vectors.
The cross product of $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ and $\begin{pmatrix} d \\ e \\ f \end{pmatrix}$ is $\begin{pmatrix} bf - ce \\ cd - af \\ ae - bd \end{pmatrix}.$
Using this formula, I get that the cross product of $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$ is $\begin{pmatrix} 2 \\ -2 \\ -1 \end{pmatrix}.$
This vector is orthogonal to both given vectors, but it is not a unit vector.
To make it a unit vector, I need to divide it by its length, which is $\sqrt{2^2 + (-2)^2 + (-1)^2} = \sqrt{9} = 3.$
Therefore, a unit vector that is orthogonal to both given vectors is $\frac{1}{3}\begin{pmatrix} 2 \\ -2 \\ -1 \end{pmatrix}.$