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1. If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

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Can a directed line have direction angles 45°, 60°, 120°?

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3. Can a directed line have direction angles 45°, 45°, 60°?

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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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5. Find the direction cosines of x, y and z-axis.

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6. Find the direction-cosines of a line which makes equal angles with the axes. How many such lines are there?

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If a line has direction ratios 2, -1, -2, determine its direction cosines.

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If a line has direction ratios 2, -1, -2, determine its direction cosines.

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9. If a line has the direction ratios – 18, 12, – 4. then what are its direction cosines?

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10. Find the direction ratios and the direction cosines of the lines joining the pairs of points:
(–2, 1, 0), (3, 2, 1).

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