If a line makes an angle of π/4 with the positive directions of each of the x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is
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Official solution
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, l2+m2+n2=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 \[{\pi \over 2}\]