QJEEM07Mad83a: JEE Main 2007 Mathematics - Class 12 JEEM07Mad83a Solution: If a line makes an angle of with the positive directions of…

Question

Verified Solution • 2025–26 Session moderate

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 :

Accepted Answer

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}

Step-by-Step Verified Solution