#### Question

Find the angle between the lines whose direction ratios are proportional to *a*, *b*, *c* and *b* − *c*, *c* − *a*, *a*− *b*.

#### Solution

\[\text{ Let } \theta \text { be the angle between the given lines } . \]

\[\text{ We have } \]

\[ a_1 = a, b_1 = b, c_1 = c \]

\[ a_2 = b - c, b_2 = c - a, c_2 = a - b\]

\[\text{ Now }, \]

\[\cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\sqrt{{a_2}^2 + {b_2}^2 + {c_2}^2}}\]

\[ = \frac{a\left( b - c \right) + b\left( c - a \right) + c\left( a - b \right)}{\sqrt{a^2 + b^2 + c^2}\sqrt{\left( b - c \right)^2 + \left( c - a \right)\left( a - b \right)}} = \frac{ab - ac + bc - ab + ac - bc}{\sqrt{a^2 + b^2 + c^2}\sqrt{\left( b - c \right)^2 + \left( c - a \right)\left( a - b \right)}} = 0\]

\[ \Rightarrow \theta = \frac{\pi}{2}\]

\[\text { Thus, the angle between the given lines measures } 90° . \]