if a line makes 60 degree angle with positive direction of x and makes 45 degree with positive direction of Y. then what is the angle made by negative direction of Z

Let the direction cosines of the line be l,m and n.
We know that
l2+m2+n2= 1.
Let the line make angle θ with the positive direction of the z-axis.
cos2α+cos2β+cos2γ=1Here α=60 and β=45 and γ= θSo cos260+cos245+cos2θ=1cos2θ=1-14-12=14cosθ=±12So θ= 60 degree or 120.and here it is given that we have to find the angle made by negative z -axisSo cosθ=-12θ=120 degree

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Let the direction cosines of the line bel,mandn.

We know thatl2+m2+n2= 1.

Let the line make angle θ with the positive direction of thez-axis.

∴ cosθ =

⇒ θ = 60 or 120º

However, it is given that this line lies in the 1stoctant. Hence, it would make a 60º angle with the positive direction of thez-axis.

∴ Angle made by the given line with the negative direction of the z-axis = 180 60 = 120

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