Three Dimensional Geometry — MATHS STD 12 Science — Question
Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSThree Dimensional Geometry3 Marks
Question
Find the direction cosines of a line which makes equal angles with the coordinate axes.
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Answer
Let a line make equal angles $\alpha,\ \alpha,\ \alpha$ with the co-ordinate axes.
$\therefore$ Direction cosines of the line are $\cos\alpha,\ \cos\alpha,\ \cos\alpha\ \ .....(\text{i})$
$\therefore\ \cos^2\alpha+\cos^2\alpha+\cos^2\alpha=1\ \ \ [\because\ \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1]$
$\Rightarrow\ 3\cos^2\alpha\ \Rightarrow\ \cos^2\alpha=\frac{1}{3}\ \Rightarrow\ \cos\alpha=\pm\frac{1}{\sqrt{3}}$
Putting $\cos\alpha=\pm\frac{1}{\sqrt{3}}$ in eq.(i), direction cosines of the required line making equal angles with the co-ordinator axes are $\pm\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}}$
Direction cosines of a line making equal angles with the co-ordinate axes in the positive i.e., first octant are $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}.$
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