Let a line passing through the point (4,1,0) intersect the line L_1: (x-1)/2=(y-2)/3=(z-3)/4 at the point A(α, β, γ) and the line L_2: x-6=y=-z+4 at…

Let a line passing through the point $\displaystyle (4,1,0)$ intersect the line $\displaystyle \mathrm{L}_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ at the point $\displaystyle A(\alpha, \beta, \gamma)$ and the line $\displaystyle \mathrm{L}_2: x-6=y=-z+4$ at the point $\displaystyle B(a, b, c)$. Then $\displaystyle \left|\begin{array}{lll}1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c\end{array}\right|$ is equal to