Two straight tunnels are being dug in a mine, and a surveyor must know whether they will break into…

Step 1. The data. $\vec{a}_2 - \vec{a}_1 = (-3, 10, 7)$.

Step 2. The cross product. $\vec{b}_1 \times \vec{b}_2 = (2 \times 1 - 3 \times (-2),\ 3 \times 3 - 1 \times 1,\ 1 \times (-2) - 2 \times 3) = (8, 8, -8)$.

Step 3. The triple product. $(-3, 10, 7) \cdot (8, 8, -8) = -24 + 80 - 56 = 0$. The tunnels are coplanar, and they are not parallel, so they meet.

Step 4. The meeting point. The $x$ and $y$ equations are $-1 + \lambda = -4 + 3\mu$ and $2\lambda = 10 - 2\mu$. The second gives $\mu = 5 - \lambda$. Then $-1 + \lambda = -4 + 15 - 3\lambda$ gives $4\lambda = 12$, so $\lambda = 3$ and $\mu = 2$.

Step 5. The check on $z$. $2 + 3 \times 3 = 11$ and $9 + 2 = 11$. They agree.