A straight cable passes through the point A(1, 2, 0) with direction ratios (4, -3, 12), with…

Step 1. The point. $2(1 + 4\lambda) - (2 - 3\lambda) + 2(12\lambda) = 35$ gives $2 + 8\lambda - 2 + 3\lambda + 24\lambda = 35$. So $35\lambda = 35$ and $\lambda = 1$. The point is $(5, -1, 12)$.

Step 2. The dot product. $\vec{b} \cdot \vec{n} = 8 + 3 + 24 = 35$.

Step 3. The lengths. $\lvert \vec{b} \rvert = \sqrt{16 + 9 + 144} = 13$ and $\lvert \vec{n} \rvert = 3$.

Step 4. The sine. $\sin\varphi = \dfrac{35}{39} \approx 0.897$, so $\varphi \approx 63.82^\circ$.