If a, b, c are real numbers that are not all equal, then the angle between the lines whose direction ratios are a, b, c and \(b-c,\ c-a,\ a-b\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{\pi}{2}\)
Since a, b, c are not all equal, neither set of direction ratios is 0, 0, 0, so both are genuine lines.
The angle depends on \(a_1a_2+b_1b_2+c_1c_2\):
\(a(b-c)+b(c-a)+c(a-b)\)
\(= ab-ac+bc-ab+ca-cb\)
\(= 0\)
The sum is zero for every such a, b, c, so \(\cos\theta = 0\) and the lines are perpendicular.