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Question
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
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Solution
\[\text{ Let } \theta \text { be the angle between the given lines } . \]
\[\text{ We have } \]
\[ a_1 = a, b_1 = b, c_1 = c \]
\[ a_2 = b - c, b_2 = c - a, c_2 = a - b\]
\[\text{ Now }, \]
\[\cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\sqrt{{a_2}^2 + {b_2}^2 + {c_2}^2}}\]
\[ = \frac{a\left( b - c \right) + b\left( c - a \right) + c\left( a - b \right)}{\sqrt{a^2 + b^2 + c^2}\sqrt{\left( b - c \right)^2 + \left( c - a \right)\left( a - b \right)}} = \frac{ab - ac + bc - ab + ac - bc}{\sqrt{a^2 + b^2 + c^2}\sqrt{\left( b - c \right)^2 + \left( c - a \right)\left( a - b \right)}} = 0\]
\[ \Rightarrow \theta = \frac{\pi}{2}\]
\[\text { Thus, the angle between the given lines measures } 90° . \]
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