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

Solution

Let l, m, n be the direction cosines of the line with direction angles 90°, 135°, 45°.
$therefore space space straight l space equals space cos space 90 degree space equals space 0 comma space space space straight m space equals space cos space 135 degree space equals space cos space left parenthesis 180 degree minus 45 degree right parenthesis space equals space minus cos space 45 degree space equals space minus fraction numerator 1 over denominator square root of 2 end fraction comma$
$straight n space equals space cos space 45 degree space equals space fraction numerator 1 over denominator square root of 2 end fraction$
$therefore space space$ direction cosines are 0, $negative fraction numerator 1 over denominator square root of 2 end fraction comma space fraction numerator 1 over denominator square root of 2 end fraction.$

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Question

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

Solution

Let l, m, n be the direction cosines of the line with direction angles 45°. 60°, 120°
∴ l = cos 45° $equals space fraction numerator 1 over denominator square root of 2 end fraction comma space space straight m space equals space cos space 60 degree space equals 1 half comma space space straight n space equals cos space 120 degree space equals space cos space left parenthesis 180 degree minus 60 degree right parenthesis space equals space minus cos space 60 degree space equals negative 1 half$
$therefore space space space straight l squared plus straight m squared plus straight n squared space equals space 1 half plus 1 fourth plus 1 fourth space equals space fraction numerator 2 plus 1 plus 1 over denominator 4 end fraction space equals 4 over 4 space equals 1$
∴ a line can have the given angles as direction angles.

Question

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

Solution

Let l, m, n be the direction cosines of the line with direction angles 45°, 45°, 60°.
$therefore space space straight l space equals cos space 45 degree space equals fraction numerator 1 over denominator square root of 2 end fraction comma space straight m space equals space cos space 45 degree space equals space fraction numerator 1 over denominator square root of 2 end fraction comma space straight n space equals space cos space 60 degree space equals space 1 half$
These values of l, m, n do not satisfy the selection l² + m² + n² = 1 as
$open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared plus space open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared plus space open parentheses 1 half close parentheses squared space equals space 1 half plus 1 half plus 1 fourth space equals 5 over 4 not equal to 1$
∴ given angles cannot be the direction angles of a line.

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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.

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.