If $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ is perpendicular to these vectors, what is $a:b:c$?

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Given that the vector \(a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\) is perpendicular to each of the vectors \(5\mathbf{i}+3\mathbf{j}-4\mathbf{k}\) and \(2\mathbf{i}-\mathbf{j}-6\mathbf{k}\), find the ratios \(a:b:c\). Hence find a unit vector which is perpendicular to each of the vectors \(5\mathbf{i}+3\mathbf{j}-4\mathbf{k}\) and \(2\mathbf{i}-\mathbf{j}-6\mathbf{k}\).