The angle between the lines whose direction ratios are \(a, b, c\) and \(b - c, c - a, a - b\) is

Let the direction ratios of the two lines be represented by vectors \(\vec{d_1}\) and \(\vec{d_2}\).

\(\vec{d_1} = a\hat{i} + b\hat{j} + c\hat{k}\)

\(\vec{d_2} = (b - c)\hat{i} + (c - a)\hat{j} + (a - b)\hat{k}\)

The angle \(\theta\) between the lines is given by \(\cos\theta = \dfrac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}| |\vec{d_2}|}\).

Calculating the dot product of the direction vectors:

\(\vec{d_1} \cdot \vec{d_2} = a(b - c) + b(c - a) + c(a - b)\)

\(\vec{d_1} \cdot \vec{d_2} = ab - ac + bc - ab + ac - bc = 0\)

Since the dot product is zero, the vectors are perpendicular.

\(\cos\theta = 0 \Rightarrow \theta = 90^\circ\)

Answer: \(90^\circ\)