### Instruction: The real numbers $a$ and $b$ satisfy
\[\begin{pmatrix} 2 \\ a \\ -7 \end{pmatrix} \times \begin{pmatrix} 5 \\ 4 \\ b \end{pmatrix} = \mathbf{0}.\]Enter the ordered pair $(a,b).$.### Response: I know that the cross product of two vectors is zero if and only if they are parallel, so I can use that property to find $a$ and $b$.
If the two vectors are parallel, then they must have the same direction, so there must be some constant $k$ such that
\[\begin{pmatrix} 5 \\ 4 \\ b \end{pmatrix} = k \begin{pmatrix} 2 \\ a \\ -7 \end{pmatrix}.\]
This implies that
\[5 = 2k, \quad 4 = ak, \quad b = -7k.\]
Solving for $k$ from the first equation, I get $k = 5/2$.
Plugging this into the second equation, I get $4 = a(5/2)$, so $a = 8/5$.
Plugging this into the third equation, I get $b = -7(5/2) = -35/2$.