Please show all work. Recall, in 3D space, direction angles alpha, beta, and gam

Question

Please show all work.

Recall, in 3D space, direction angles alpha, beta, and gamma are angles measured from a vector to the x,y and z axis, respectively. Is it possible for a 3D vector to be drawn with direction angles of alpha =45 degree and beta =45 degree ? If yes, what is the measure for the direction angle gamma ? Is it possible for a 3D vector to be drawn with direction angles of alpha =40 degree and beta =40 degree ? If yes, what is the measure of the direction angle gamma ? Is it possible for a 3D vector to be drawn with direction angles of alpha = 50 degree and beta =50 degree ? If yes, what is the measure of the direction angle gamma ?

Explanation / Answer

a) cos^2 (alpha) + cos^2 (beta) + cos^2 (gamma) = 1 cos^2(45) + cos^2 (45) +cos^2 (gamma) =1 =>0.5 + 0.5 +cos^2 (gamma)=1 => cos^2 (gamma)= 0 => gamma= 90 degree b) cos^2 (alpha) + cos^2 (beta) + cos^2 (gamma) = 1 cos^2(40) + cos^2 (40) +cos^2 (gamma) =1 =>0.5868 + 0.5868 +cos^2 (gamma)=1 => cos^2 (gamma)= -0.1736 => gamma= 100 degree c) cos^2 (alpha) + cos^2 (beta) + cos^2 (gamma) = 1 cos^2(50) + cos^2 (50) +cos^2 (gamma) =1 =>0.4131 + 0.4131 +cos^2 (gamma)=1 => cos^2 (gamma)= 0.1736 => gamma= 80 degree

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