JEE MAIN - Mathematics (2007 - No. 6)
If a line makes an angle of $$\pi /4$$ with the positive directions of each of $$x$$-axis and $$y$$-axis, then the angle that the line makes with the positive direction of the $$z$$-axis is :
$${\pi \over 4}$$
$${\pi \over 2}$$
$${\pi \over 6}$$
$${\pi \over 3}$$
Explanation
Let the angle of line makes with the positive direction of $$z$$-axis is $$\alpha $$ direction cosines of line with the $$+ve$$ directions of $$x$$-axis, $$y$$-axis, and $$z$$-axis is $$l,$$ $$m,$$ $$n$$ respectively.
$$\therefore$$ $$l = \cos {\pi \over 4},m = \cos {\pi \over 4},\,\,n = cos\,\alpha $$
as we know that, $${l^2} + {m^2} + {n^2} = 1$$
$$\therefore$$ $${\cos ^2}{\pi \over 4} + {\cos ^2}{\pi \over 4} + {\cos ^2}\alpha = 1$$
$$ \Rightarrow {1 \over 2} + {1 \over 2} + {\cos ^2}\alpha = 1$$
$$ \Rightarrow {\cos ^2}\alpha = 0 \Rightarrow \alpha = {\pi \over 2}$$
Hence, angle with positive direction of the $$z$$-axis is $${\pi \over 2}$$
$$\therefore$$ $$l = \cos {\pi \over 4},m = \cos {\pi \over 4},\,\,n = cos\,\alpha $$
as we know that, $${l^2} + {m^2} + {n^2} = 1$$
$$\therefore$$ $${\cos ^2}{\pi \over 4} + {\cos ^2}{\pi \over 4} + {\cos ^2}\alpha = 1$$
$$ \Rightarrow {1 \over 2} + {1 \over 2} + {\cos ^2}\alpha = 1$$
$$ \Rightarrow {\cos ^2}\alpha = 0 \Rightarrow \alpha = {\pi \over 2}$$
Hence, angle with positive direction of the $$z$$-axis is $${\pi \over 2}$$
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