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Question
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
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Solution
\[ \text{ We know that if a line has direction ratio } \left( a, b, c \right), \text{ then the cosine of its angle with the z - axis is given } by\]
\[\cos \gamma = \frac{c}{\sqrt{a^2 + b^2 + c^2}}\]
\[\text { Suppose the inclination of the line with direction ratio } \left( 0, 1, - 1 \right) \text{ with z - axis is } \gamma . \]
\[\text{ Now }, \]
\[\cos \lambda = \frac{- 1}{\sqrt{0 + 1 + 1}}\]
\[ = - \frac{1}{\sqrt{2}} \]
\[ \Rightarrow \lambda = \frac{3\pi}{4}\]
\[\text{ Hence, the inclination of the line with z - axis is } \frac{3\pi}{4} . \]
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