If the direction ratios of two lines are \(a, b, c\) and \((b-c), (c-a), (a - b)\) respectively, then the angle between these lines is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{\pi}{2}\)
The angle \(\theta\) between lines with direction ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) has \(\cos\theta\) proportional to \(a_1a_2 + b_1b_2 + c_1c_2\), so the lines are perpendicular exactly when this sum is 0.
\(a(b - c) + b(c - a) + c(a - b)\)
\(= ab - ac + bc - ab + ac - bc\)
\(= 0\)
So \(\cos\theta = 0\) and the angle between the lines is \(\frac{\pi}{2}\).
NCERT (2026-27 edition):
Three Dimensional Geometry
