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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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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.

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

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Let α be the angle which the line with all the axes,
∴ its direction-cosines are cos α, cos α, cos α

therefore space space cos squared straight alpha plus cos squared straight alpha plus cos squared straight alpha space equals space 1

open square brackets because space space straight l squared plus straight m squared plus straight n squared space equals space 1 close square brackets

therefore space space space 3 space cos space squared space equals space 1 comma space space space space or space space space cos squared straight alpha space equals space 1 third

therefore space space space space space space cos space straight alpha space equals space plus-or-minus space fraction numerator 1 over denominator square root of 3 end fraction

therefore

required direction-cosines are

plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction comma space plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction comma space plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction

These are four different groups of signs
i.e., +, +, +
+, -, +
+, +, -
+, -, -
∴ there are four distinct lines.

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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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