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Question
Cartesian equations of a line AB are \[\frac{2x - 1}{2} = \frac{4 - y}{7} = \frac{z + 1}{2} .\] Write the direction ratios of a line parallel to AB.
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Solution
We have ,
\[\frac{2x - 1}{2} = \frac{4 - y}{7} = \frac{z + 1}{2}\]
The equation of the line AB can be re-written as ,
\[\frac{x - \frac{1}{2}}{1} = \frac{y - 4}{- 7} = \frac{z + 1}{2}\]
The direction ratios of the line parallel to AB are proportional to 1, -7, 2 .
Also, the direction cosines of the line parallel to AB are proportional to
\[\frac{1}{\sqrt{1^2 + \left( - 7 \right)^2 + 2^2}}, \frac{- 7}{\sqrt{1^2 + \left( - 7 \right)^2 + 2^2}}, \frac{2}{\sqrt{1^2 + \left( - 7 \right)^2 + 2^2}} \]
\[ = \frac{1}{\sqrt{54}}, \frac{- 7}{\sqrt{54}}, \frac{2}{\sqrt{54}}\]
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