However, cross product outputs are orthogonal to $\mathbf{a}$. Check:
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- z + 4y = -1
- \Rightarrow \text{Subtract: } 0 = 6 \Rightarrow \text{Contradiction}
- This suggests no solution exists unless the given cross product is inconsistent with the vector.
- \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} = 10 + 6 - 4 = 12 \ne 0
- So the given vector is not orthogonal to $\mathbf{a}$, which violates the cross product property.
- Therefore, no such vector $\mathbf{v}$ exists.
