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Three Dimensional Geometry

Question

A line makes an angle of $straight pi over 4$ with each of y-axis and z-axis. Find the angle made by it with x-axis.

Solution

Let a be the angle which the line makes with positive axis of x.

therefore

direction cosines of the line are

cos space straight alpha comma space space cos space straight pi over 4 comma space space cos straight pi over 4

therefore space space space cos squared space straight alpha space plus space cos space squared space straight pi over 4 space plus space cos squared straight pi over 4 space equals space 1

open square brackets cos squared straight alpha plus cos squared straight beta plus cos squared straight gamma space equals space 1 close square brackets

therefore

cos squared straight alpha plus 1 half plus 1 half space equals space 1 space space space space rightwards double arrow space space space space cos squared straight alpha space equals space 0 space space space rightwards double arrow space space space space cos space straight alpha space equals space 0 space space space space space rightwards double arrow space space space straight a space equals space straight pi over 2

∴ line makes a right angle with x-axis.

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