The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\begin{pmatrix} \frac{1}{50} & \frac{7}{50} \\ \frac{7}{50} & \frac{49}{50} \end{pmatrix}.\]Find the direction vector of line $\ell.$  Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix},$ where $a,$ and $b$ are integers, $a > 0,$ and $\gcd(|a|,|b|) = 1.$
Let $\mathbf{P}$ denote the given matrix, so $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto $\ell.$  In particular, $\mathbf{P} \mathbf{v}$ lies on $\ell$ for any vector $\mathbf{v}.$  So, we can take $\mathbf{v} = \mathbf{i}.$  Then
\[\mathbf{P} \mathbf{i} = \begin{pmatrix} \frac{1}{50} \\ \frac{7}{50} \end{pmatrix} = \frac{1}{50} \begin{pmatrix} 1 \\ 7 \end{pmatrix}.\]Thus, the direction vector we seek is $\boxed{\begin{pmatrix} 1 \\ 7 \end{pmatrix}}.$