Four joints of a steel frame have the coordinates A(1, 2, 3), B(2, 4, 5), C(0, 1, 1) and D(2, 2…

Step 1. The vectors. $\overrightarrow{AB} = (1, 2, 2)$, $\overrightarrow{AC} = (-1, -1, -2)$ and $\overrightarrow{AD} = (1, 0, \lambda - 3)$.

Step 2. The determinant. $

= 1\big[-(\lambda - 3) - 0\big] - 2\big[-(\lambda - 3) + 2\big] + 2\big[0 + 1\big]$.

Step 3. Simplify. $-(\lambda - 3) + 2(\lambda - 3) - 4 + 2 = \lambda - 3 - 2 = \lambda - 5$.

Step 4. Solve. $\lambda - 5 = 0$, so $\lambda = 5$.