Let $\mathbf{v}_1,$ $\mathbf{v}_2,$ $\dots,$ $\mathbf{v}_k$ be unit vectors such that
\[\mathbf{v}_1 + \mathbf{v}_2 + \dots + \mathbf{v}_k = \begin{pmatrix} 6 \\ -5 \end{pmatrix}.\]Find the smallest possible value of $k.$

Note: A unit vector is a vector of magnitude 1.
I need to find the smallest number of unit vectors that can add up to a given vector. One way to approach this problem is to use the triangle inequality, which states that the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side. If I apply the triangle inequality to the unit vectors and the given vector, I get that
\[\left\|\mathbf{v}_1 + \mathbf{v}_2 + \dots + \mathbf{v}_k\right\| \leq \left\|\mathbf{v}_1\right\| + \left\|\mathbf{v}_2\right\| + \dots + \left\|\mathbf{v}_k\right\|.\] Since the unit vectors have magnitude 1, this simplifies to
\[\left\|\mathbf{v}_1 + \mathbf{v}_2 + \dots + \mathbf{v}_k\right\| \leq k.\] Now, I need to find the magnitude of the given vector, which is the square root of the sum of the squares of its components. I have
\[\left\|\begin{pmatrix} 6 \\ -5 \end{pmatrix}\right\| = \sqrt{6^2 + (-5)^2} = \sqrt{61}.\] Therefore, I have the inequality
\[\sqrt{61} \leq k.\] To find the smallest possible value of $k,$ I need to round up $\sqrt{61}$ to the next integer, since $k$ has to be a positive integer. I can use a calculator or an approximation to find that $\sqrt{61}$ is about 7.81. Rounding up, I get that the smallest possible value of $k$ is 8. This means that I need at least 8 unit vectors to add up to the given vector. 