How to show that a set of vectors do not span $\mathbb R^3$? How do I show that these vectors $\{(-1, -2, -1), (-1,-5,-1), (-1,-1,-1)\}$ do not span in $\mathbb{R}^3$ by giving a vector not in there span? 
 A: Notice that all the vectors have the same first and last components. Therefore any linear combination will also have the same first and last components. So choose a vector with different ones, like $(1,0,0)$.
A: Here are a couple of more general ways you can use in other cases where an orthogonal vector can't be found immediately (a la Chappers' answer):
Method You Can Use in $\Bbb R^n$ for Any $n$:


*

*Verify your vectors are linearly dependent.  You can do so using Gauss-Jordan elimination.

*If they are not linearly independent, throw out any vectors that are a linear combination of the others.  NOTE: Obviously none of the three vectors in this case are multiples of the others so the maximum you will need to throw out is $1$ vector.

*Construct a matrix with the remaining vectors as rows.

*Solve the homogeneous system to obtain the set of vectors that is orthogonal to your set.


Method You Can Use Because Your Vectors Happen to Be in $\Bbb R^3$ Where the Cross Product Is Defined:


*

*To verify your vectors are linearly dependent you could take the triple scalar product.  If it is zero, the vectors are linearly dependent.

*Same as above.

*Take the cross product of the two remaining vectors to obtain a vector orthogonal to your set.

